Properties

Label 2-370-185.68-c1-0-5
Degree $2$
Conductor $370$
Sign $0.985 + 0.172i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−2.09 + 2.09i)3-s + 4-s + (−1.81 − 1.30i)5-s + (2.09 − 2.09i)6-s + (−0.643 + 0.643i)7-s − 8-s − 5.74i·9-s + (1.81 + 1.30i)10-s − 1.88i·11-s + (−2.09 + 2.09i)12-s − 0.536·13-s + (0.643 − 0.643i)14-s + (6.52 − 1.06i)15-s + 16-s − 0.334i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−1.20 + 1.20i)3-s + 0.5·4-s + (−0.812 − 0.583i)5-s + (0.853 − 0.853i)6-s + (−0.243 + 0.243i)7-s − 0.353·8-s − 1.91i·9-s + (0.574 + 0.412i)10-s − 0.568i·11-s + (−0.603 + 0.603i)12-s − 0.148·13-s + (0.172 − 0.172i)14-s + (1.68 − 0.276i)15-s + 0.250·16-s − 0.0811i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.985 + 0.172i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440305 - 0.0382533i\)
\(L(\frac12)\) \(\approx\) \(0.440305 - 0.0382533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (1.81 + 1.30i)T \)
37 \( 1 + (0.819 + 6.02i)T \)
good3 \( 1 + (2.09 - 2.09i)T - 3iT^{2} \)
7 \( 1 + (0.643 - 0.643i)T - 7iT^{2} \)
11 \( 1 + 1.88iT - 11T^{2} \)
13 \( 1 + 0.536T + 13T^{2} \)
17 \( 1 + 0.334iT - 17T^{2} \)
19 \( 1 + (-1.08 - 1.08i)T + 19iT^{2} \)
23 \( 1 - 4.65T + 23T^{2} \)
29 \( 1 + (-3.33 + 3.33i)T - 29iT^{2} \)
31 \( 1 + (-4.90 - 4.90i)T + 31iT^{2} \)
41 \( 1 + 8.26iT - 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + (0.141 - 0.141i)T - 47iT^{2} \)
53 \( 1 + (5.03 + 5.03i)T + 53iT^{2} \)
59 \( 1 + (4.10 + 4.10i)T + 59iT^{2} \)
61 \( 1 + (8.90 + 8.90i)T + 61iT^{2} \)
67 \( 1 + (3.61 + 3.61i)T + 67iT^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + (-6.82 + 6.82i)T - 73iT^{2} \)
79 \( 1 + (5.73 + 5.73i)T + 79iT^{2} \)
83 \( 1 + (7.90 + 7.90i)T + 83iT^{2} \)
89 \( 1 + (-1.21 + 1.21i)T - 89iT^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08674327754436736374384312279, −10.65721877857478613843386316398, −9.503729607199426442347771825936, −8.922148175345323080149892951194, −7.75651403275331640771401260637, −6.46711874411291566538308792947, −5.45914695603766580820278398404, −4.52696074741094433816077862309, −3.33400603416411901869419434893, −0.57860011343253862190412420651, 1.01277652586633846204674401927, 2.75114026651608260016301103032, 4.63744220605933985362135059831, 6.05859075082357513176134393802, 6.91418333534597266937073185985, 7.39339464363610729478851461693, 8.318447776456928850359093802706, 9.792692366510732398202086532231, 10.79144988425503147936755535915, 11.36934427652155902290465001675

Graph of the $Z$-function along the critical line