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Properties

Label 2-245-245.103-c1-0-10
Degree 22
Conductor 245245
Sign 0.988+0.148i0.988 + 0.148i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 + 1.68i)2-s + (−2.33 + 0.441i)3-s + (−0.697 − 2.26i)4-s + (−2.23 − 0.156i)5-s + (2.15 − 4.46i)6-s + (−1.67 + 2.04i)7-s + (0.724 + 0.253i)8-s + (2.44 − 0.961i)9-s + (3.03 − 3.55i)10-s + (0.785 − 2.00i)11-s + (2.62 + 4.96i)12-s + (4.20 + 0.473i)13-s + (−1.35 − 5.36i)14-s + (5.26 − 0.619i)15-s + (2.58 − 1.76i)16-s + (−1.85 − 0.0695i)17-s + ⋯
L(s)  = 1  + (−0.877 + 1.18i)2-s + (−1.34 + 0.254i)3-s + (−0.348 − 1.13i)4-s + (−0.997 − 0.0700i)5-s + (0.878 − 1.82i)6-s + (−0.634 + 0.773i)7-s + (0.256 + 0.0896i)8-s + (0.816 − 0.320i)9-s + (0.958 − 1.12i)10-s + (0.236 − 0.603i)11-s + (0.757 + 1.43i)12-s + (1.16 + 0.131i)13-s + (−0.362 − 1.43i)14-s + (1.36 − 0.159i)15-s + (0.646 − 0.440i)16-s + (−0.450 − 0.0168i)17-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=((0.988+0.148i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+1/2) L(s)=((0.988+0.148i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: 0.988+0.148i0.988 + 0.148i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ245(103,⋅)\chi_{245} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 245, ( :1/2), 0.988+0.148i)(2,\ 245,\ (\ :1/2),\ 0.988 + 0.148i)

Particular Values

L(1)L(1) ≈\approx 0.167440−0.0124973i0.167440 - 0.0124973i
L(12)L(\frac12) ≈\approx 0.167440−0.0124973i0.167440 - 0.0124973i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(2.23+0.156i)T 1 + (2.23 + 0.156i)T
7 1+(1.67−2.04i)T 1 + (1.67 - 2.04i)T
good2 1+(1.24−1.68i)T+(−0.589−1.91i)T2 1 + (1.24 - 1.68i)T + (-0.589 - 1.91i)T^{2}
3 1+(2.33−0.441i)T+(2.79−1.09i)T2 1 + (2.33 - 0.441i)T + (2.79 - 1.09i)T^{2}
11 1+(−0.785+2.00i)T+(−8.06−7.48i)T2 1 + (-0.785 + 2.00i)T + (-8.06 - 7.48i)T^{2}
13 1+(−4.20−0.473i)T+(12.6+2.89i)T2 1 + (-4.20 - 0.473i)T + (12.6 + 2.89i)T^{2}
17 1+(1.85+0.0695i)T+(16.9+1.27i)T2 1 + (1.85 + 0.0695i)T + (16.9 + 1.27i)T^{2}
19 1+(3.30−5.72i)T+(−9.5−16.4i)T2 1 + (3.30 - 5.72i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.262+7.01i)T+(−22.9+1.71i)T2 1 + (0.262 + 7.01i)T + (-22.9 + 1.71i)T^{2}
29 1+(−4.54+1.03i)T+(26.1−12.5i)T2 1 + (-4.54 + 1.03i)T + (26.1 - 12.5i)T^{2}
31 1+(3.46−1.99i)T+(15.5−26.8i)T2 1 + (3.46 - 1.99i)T + (15.5 - 26.8i)T^{2}
37 1+(0.342−0.180i)T+(20.8−30.5i)T2 1 + (0.342 - 0.180i)T + (20.8 - 30.5i)T^{2}
41 1+(4.04+8.40i)T+(−25.5+32.0i)T2 1 + (4.04 + 8.40i)T + (-25.5 + 32.0i)T^{2}
43 1+(3.00+8.58i)T+(−33.6+26.8i)T2 1 + (3.00 + 8.58i)T + (-33.6 + 26.8i)T^{2}
47 1+(4.09+3.02i)T+(13.8+44.9i)T2 1 + (4.09 + 3.02i)T + (13.8 + 44.9i)T^{2}
53 1+(−0.148−0.0786i)T+(29.8+43.7i)T2 1 + (-0.148 - 0.0786i)T + (29.8 + 43.7i)T^{2}
59 1+(0.00101+0.0136i)T+(−58.3+8.79i)T2 1 + (0.00101 + 0.0136i)T + (-58.3 + 8.79i)T^{2}
61 1+(−3.58+11.6i)T+(−50.4−34.3i)T2 1 + (-3.58 + 11.6i)T + (-50.4 - 34.3i)T^{2}
67 1+(−4.88−1.30i)T+(58.0+33.5i)T2 1 + (-4.88 - 1.30i)T + (58.0 + 33.5i)T^{2}
71 1+(1.72−7.57i)T+(−63.9−30.8i)T2 1 + (1.72 - 7.57i)T + (-63.9 - 30.8i)T^{2}
73 1+(−5.89+4.35i)T+(21.5−69.7i)T2 1 + (-5.89 + 4.35i)T + (21.5 - 69.7i)T^{2}
79 1+(5.78+3.34i)T+(39.5+68.4i)T2 1 + (5.78 + 3.34i)T + (39.5 + 68.4i)T^{2}
83 1+(−1.34−11.9i)T+(−80.9+18.4i)T2 1 + (-1.34 - 11.9i)T + (-80.9 + 18.4i)T^{2}
89 1+(0.169+0.432i)T+(−65.2+60.5i)T2 1 + (0.169 + 0.432i)T + (-65.2 + 60.5i)T^{2}
97 1+(7.26+7.26i)T+97iT2 1 + (7.26 + 7.26i)T + 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.01601403354931607980252340861, −10.99012188085329801004095372779, −10.17535894864725728885321347251, −8.633267320177946730715587689513, −8.444086676529992751830558740710, −6.74092184217255592045190334345, −6.28841172517983115895199281692, −5.31985683804018998724881762217, −3.74890257929561437057362229798, −0.26084742556146350141059789766, 1.10117821158153632838409199031, 3.29318054435875218983654955378, 4.52390321764828513882375428011, 6.24761809132415478005788495766, 7.12967709370191621825938582707, 8.406216326411728750112268958091, 9.537610439515665619673990661769, 10.58040981864368628479599123106, 11.24187153653947568658615899360, 11.60286666625971340940824397861

Graph of the ZZ-function along the critical line