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Properties

Label 2-380-19.16-c1-0-5
Degree 22
Conductor 380380
Sign −0.956+0.290i-0.956 + 0.290i
Analytic cond. 3.034313.03431
Root an. cond. 1.741921.74192
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  + (−0.693 + 0.252i)3-s + (−0.173 + 0.984i)5-s + (−2.32 − 4.03i)7-s + (−1.88 + 1.57i)9-s + (−1.20 + 2.09i)11-s + (−2.11 − 0.768i)13-s + (−0.128 − 0.726i)15-s + (−0.901 − 0.756i)17-s + (−3.98 − 1.77i)19-s + (2.63 + 2.20i)21-s + (−0.255 − 1.44i)23-s + (−0.939 − 0.342i)25-s + (2.01 − 3.48i)27-s + (−3.72 + 3.12i)29-s + (−3.60 − 6.24i)31-s + ⋯
L(s)  = 1  + (−0.400 + 0.145i)3-s + (−0.0776 + 0.440i)5-s + (−0.880 − 1.52i)7-s + (−0.626 + 0.526i)9-s + (−0.364 + 0.630i)11-s + (−0.585 − 0.213i)13-s + (−0.0330 − 0.187i)15-s + (−0.218 − 0.183i)17-s + (−0.913 − 0.406i)19-s + (0.574 + 0.482i)21-s + (−0.0532 − 0.301i)23-s + (−0.187 − 0.0684i)25-s + (0.387 − 0.671i)27-s + (−0.691 + 0.580i)29-s + (−0.647 − 1.12i)31-s + ⋯

Functional equation

Λ(s)=(380s/2 ΓC(s) L(s)=((−0.956+0.290i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(380s/2 ΓC(s+1/2) L(s)=((−0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 380380    =    22⋅5⋅192^{2} \cdot 5 \cdot 19
Sign: −0.956+0.290i-0.956 + 0.290i
Analytic conductor: 3.034313.03431
Root analytic conductor: 1.741921.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ380(301,⋅)\chi_{380} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 380, ( :1/2), −0.956+0.290i)(2,\ 380,\ (\ :1/2),\ -0.956 + 0.290i)

Particular Values

L(1)L(1) ≈\approx 0.0177552−0.119664i0.0177552 - 0.119664i
L(12)L(\frac12) ≈\approx 0.0177552−0.119664i0.0177552 - 0.119664i
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)
bad2 1 1
5 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
19 1+(3.98+1.77i)T 1 + (3.98 + 1.77i)T
good3 1+(0.693−0.252i)T+(2.29−1.92i)T2 1 + (0.693 - 0.252i)T + (2.29 - 1.92i)T^{2}
7 1+(2.32+4.03i)T+(−3.5+6.06i)T2 1 + (2.32 + 4.03i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.20−2.09i)T+(−5.5−9.52i)T2 1 + (1.20 - 2.09i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.11+0.768i)T+(9.95+8.35i)T2 1 + (2.11 + 0.768i)T + (9.95 + 8.35i)T^{2}
17 1+(0.901+0.756i)T+(2.95+16.7i)T2 1 + (0.901 + 0.756i)T + (2.95 + 16.7i)T^{2}
23 1+(0.255+1.44i)T+(−21.6+7.86i)T2 1 + (0.255 + 1.44i)T + (-21.6 + 7.86i)T^{2}
29 1+(3.72−3.12i)T+(5.03−28.5i)T2 1 + (3.72 - 3.12i)T + (5.03 - 28.5i)T^{2}
31 1+(3.60+6.24i)T+(−15.5+26.8i)T2 1 + (3.60 + 6.24i)T + (-15.5 + 26.8i)T^{2}
37 1−2.81T+37T2 1 - 2.81T + 37T^{2}
41 1+(5.05−1.84i)T+(31.4−26.3i)T2 1 + (5.05 - 1.84i)T + (31.4 - 26.3i)T^{2}
43 1+(1.33−7.59i)T+(−40.4−14.7i)T2 1 + (1.33 - 7.59i)T + (-40.4 - 14.7i)T^{2}
47 1+(−4.70+3.94i)T+(8.16−46.2i)T2 1 + (-4.70 + 3.94i)T + (8.16 - 46.2i)T^{2}
53 1+(−0.390−2.21i)T+(−49.8+18.1i)T2 1 + (-0.390 - 2.21i)T + (-49.8 + 18.1i)T^{2}
59 1+(−8.78−7.36i)T+(10.2+58.1i)T2 1 + (-8.78 - 7.36i)T + (10.2 + 58.1i)T^{2}
61 1+(0.285+1.62i)T+(−57.3+20.8i)T2 1 + (0.285 + 1.62i)T + (-57.3 + 20.8i)T^{2}
67 1+(−1.80+1.51i)T+(11.6−65.9i)T2 1 + (-1.80 + 1.51i)T + (11.6 - 65.9i)T^{2}
71 1+(−1.11+6.29i)T+(−66.7−24.2i)T2 1 + (-1.11 + 6.29i)T + (-66.7 - 24.2i)T^{2}
73 1+(−3.18+1.15i)T+(55.9−46.9i)T2 1 + (-3.18 + 1.15i)T + (55.9 - 46.9i)T^{2}
79 1+(−2.11+0.770i)T+(60.5−50.7i)T2 1 + (-2.11 + 0.770i)T + (60.5 - 50.7i)T^{2}
83 1+(5.11+8.85i)T+(−41.5+71.8i)T2 1 + (5.11 + 8.85i)T + (-41.5 + 71.8i)T^{2}
89 1+(−5.26−1.91i)T+(68.1+57.2i)T2 1 + (-5.26 - 1.91i)T + (68.1 + 57.2i)T^{2}
97 1+(−4.19−3.52i)T+(16.8+95.5i)T2 1 + (-4.19 - 3.52i)T + (16.8 + 95.5i)T^{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

−10.71604992291978735786704847730, −10.31362819800979881614185158886, −9.340498911412799031802533290974, −7.88758641854511586274670860888, −7.13902071408443079066153677310, −6.26906859345727498107098487874, −4.93974866310573931738343063428, −3.90257261300548448075609762739, −2.54160815415552779591803528076, −0.07761952621198305518721249874, 2.33485656548037077746275115247, 3.58612588941893841621277003151, 5.30128442164485013303558407429, 5.88189852552313329998608773840, 6.81656482733590780671920418022, 8.371454291877661228307533513195, 8.926284968727498448994716020691, 9.795833117645334082849209755845, 11.03848935723972375456116410603, 11.94802192613208101474229473289

Graph of the ZZ-function along the critical line