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Properties

Label 2-750-25.11-c1-0-13
Degree 22
Conductor 750750
Sign −0.187+0.982i-0.187 + 0.982i
Analytic cond. 5.988785.98878
Root an. cond. 2.447192.44719
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)6-s + 1.72·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (1.97 − 1.43i)11-s + (−0.809 − 0.587i)12-s + (−2.40 − 1.74i)13-s + (1.39 − 1.01i)14-s + (−0.809 − 0.587i)16-s + (−1.03 − 3.18i)17-s − 0.999·18-s + (0.694 + 2.13i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.126 − 0.388i)6-s + 0.652·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (0.595 − 0.432i)11-s + (−0.233 − 0.169i)12-s + (−0.666 − 0.484i)13-s + (0.373 − 0.271i)14-s + (−0.202 − 0.146i)16-s + (−0.250 − 0.771i)17-s − 0.235·18-s + (0.159 + 0.490i)19-s + ⋯

Functional equation

Λ(s)=(750s/2 ΓC(s) L(s)=((−0.187+0.982i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(750s/2 ΓC(s+1/2) L(s)=((−0.187+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 750750    =    2⋅3⋅532 \cdot 3 \cdot 5^{3}
Sign: −0.187+0.982i-0.187 + 0.982i
Analytic conductor: 5.988785.98878
Root analytic conductor: 2.447192.44719
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ750(301,⋅)\chi_{750} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 750, ( :1/2), −0.187+0.982i)(2,\ 750,\ (\ :1/2),\ -0.187 + 0.982i)

Particular Values

L(1)L(1) ≈\approx 1.46554−1.77153i1.46554 - 1.77153i
L(12)L(\frac12) ≈\approx 1.46554−1.77153i1.46554 - 1.77153i
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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
3 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
5 1 1
good7 1−1.72T+7T2 1 - 1.72T + 7T^{2}
11 1+(−1.97+1.43i)T+(3.39−10.4i)T2 1 + (-1.97 + 1.43i)T + (3.39 - 10.4i)T^{2}
13 1+(2.40+1.74i)T+(4.01+12.3i)T2 1 + (2.40 + 1.74i)T + (4.01 + 12.3i)T^{2}
17 1+(1.03+3.18i)T+(−13.7+9.99i)T2 1 + (1.03 + 3.18i)T + (-13.7 + 9.99i)T^{2}
19 1+(−0.694−2.13i)T+(−15.3+11.1i)T2 1 + (-0.694 - 2.13i)T + (-15.3 + 11.1i)T^{2}
23 1+(−7.37+5.35i)T+(7.10−21.8i)T2 1 + (-7.37 + 5.35i)T + (7.10 - 21.8i)T^{2}
29 1+(2.89−8.91i)T+(−23.4−17.0i)T2 1 + (2.89 - 8.91i)T + (-23.4 - 17.0i)T^{2}
31 1+(1.89+5.82i)T+(−25.0+18.2i)T2 1 + (1.89 + 5.82i)T + (-25.0 + 18.2i)T^{2}
37 1+(−2.67−1.94i)T+(11.4+35.1i)T2 1 + (-2.67 - 1.94i)T + (11.4 + 35.1i)T^{2}
41 1+(−4.95−3.60i)T+(12.6+38.9i)T2 1 + (-4.95 - 3.60i)T + (12.6 + 38.9i)T^{2}
43 1+7.06T+43T2 1 + 7.06T + 43T^{2}
47 1+(−1.54+4.74i)T+(−38.0−27.6i)T2 1 + (-1.54 + 4.74i)T + (-38.0 - 27.6i)T^{2}
53 1+(−2.11+6.51i)T+(−42.8−31.1i)T2 1 + (-2.11 + 6.51i)T + (-42.8 - 31.1i)T^{2}
59 1+(−0.147−0.107i)T+(18.2+56.1i)T2 1 + (-0.147 - 0.107i)T + (18.2 + 56.1i)T^{2}
61 1+(4.16−3.02i)T+(18.8−58.0i)T2 1 + (4.16 - 3.02i)T + (18.8 - 58.0i)T^{2}
67 1+(−3.67−11.3i)T+(−54.2+39.3i)T2 1 + (-3.67 - 11.3i)T + (-54.2 + 39.3i)T^{2}
71 1+(3.51−10.8i)T+(−57.4−41.7i)T2 1 + (3.51 - 10.8i)T + (-57.4 - 41.7i)T^{2}
73 1+(−5.44+3.95i)T+(22.5−69.4i)T2 1 + (-5.44 + 3.95i)T + (22.5 - 69.4i)T^{2}
79 1+(3.50−10.8i)T+(−63.9−46.4i)T2 1 + (3.50 - 10.8i)T + (-63.9 - 46.4i)T^{2}
83 1+(−2.82−8.69i)T+(−67.1+48.7i)T2 1 + (-2.82 - 8.69i)T + (-67.1 + 48.7i)T^{2}
89 1+(−6.56+4.77i)T+(27.5−84.6i)T2 1 + (-6.56 + 4.77i)T + (27.5 - 84.6i)T^{2}
97 1+(−3.05+9.40i)T+(−78.4−57.0i)T2 1 + (-3.05 + 9.40i)T + (-78.4 - 57.0i)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

−10.26499253892338593818216072331, −9.239217839085251635111737943729, −8.436227246887676208731192774841, −7.37649734424147841180067063931, −6.63907253246798419157100698411, −5.47097884642711357925774555429, −4.70651104862278520056304430005, −3.41056128865499857893186839027, −2.39432410281391021779316309777, −1.04075997064099875422890647544, 1.91543962907373329555912867210, 3.31025095032694638901118378278, 4.39391060413323703954393623504, 4.99302696164616838846419719866, 6.09116777925122335226808004826, 7.14902774454321612704066274254, 7.85721371860222511979912960455, 9.000168447081126891459849708932, 9.511346983118703950485625370638, 10.75703850510693853819014125203

Graph of the ZZ-function along the critical line