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Properties

Label 2-370-185.117-c1-0-14
Degree $2$
Conductor $370$
Sign $-0.995 - 0.0961i$
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 + (−1.23 − 1.23i)3-s + 4-s + (−1.26 − 1.84i)5-s + (1.23 + 1.23i)6-s + (1.28 + 1.28i)7-s − 8-s + 0.0591i·9-s + (1.26 + 1.84i)10-s − 4.27i·11-s + (−1.23 − 1.23i)12-s − 1.50·13-s + (−1.28 − 1.28i)14-s + (−0.708 + 3.84i)15-s + 16-s + 2.12i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.714 − 0.714i)3-s + 0.5·4-s + (−0.567 − 0.823i)5-s + (0.504 + 0.504i)6-s + (0.486 + 0.486i)7-s − 0.353·8-s + 0.0197i·9-s + (0.401 + 0.582i)10-s − 1.28i·11-s + (−0.357 − 0.357i)12-s − 0.418·13-s + (−0.343 − 0.343i)14-s + (−0.182 + 0.993i)15-s + 0.250·16-s + 0.514i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.995 - 0.0961i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.995 - 0.0961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0159008 + 0.330015i\)
\(L(\frac12)\) \(\approx\) \(0.0159008 + 0.330015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 1.84i)T \)
37 \( 1 + (5.63 + 2.28i)T \)
good3 \( 1 + (1.23 + 1.23i)T + 3iT^{2} \)
7 \( 1 + (-1.28 - 1.28i)T + 7iT^{2} \)
11 \( 1 + 4.27iT - 11T^{2} \)
13 \( 1 + 1.50T + 13T^{2} \)
17 \( 1 - 2.12iT - 17T^{2} \)
19 \( 1 + (0.381 - 0.381i)T - 19iT^{2} \)
23 \( 1 + 8.90T + 23T^{2} \)
29 \( 1 + (-0.279 - 0.279i)T + 29iT^{2} \)
31 \( 1 + (1.59 - 1.59i)T - 31iT^{2} \)
41 \( 1 - 0.270iT - 41T^{2} \)
43 \( 1 - 0.302T + 43T^{2} \)
47 \( 1 + (3.17 + 3.17i)T + 47iT^{2} \)
53 \( 1 + (-9.71 + 9.71i)T - 53iT^{2} \)
59 \( 1 + (4.06 - 4.06i)T - 59iT^{2} \)
61 \( 1 + (-4.47 + 4.47i)T - 61iT^{2} \)
67 \( 1 + (6.17 - 6.17i)T - 67iT^{2} \)
71 \( 1 + 7.90T + 71T^{2} \)
73 \( 1 + (-8.32 - 8.32i)T + 73iT^{2} \)
79 \( 1 + (2.20 - 2.20i)T - 79iT^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 + (-5.24 - 5.24i)T + 89iT^{2} \)
97 \( 1 + 12.2iT - 97T^{2} \)
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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

−11.18258709526931217031690269929, −10.02219003187321127337467000362, −8.737563948130830268558952456921, −8.299194325665426695882212311653, −7.27366994226183147917517545482, −6.08834691295430008122030074110, −5.36112198198438168845403131649, −3.72680851803897255429906924026, −1.75549983936233523907562518504, −0.29745285762675828136349118838, 2.22615220952263096630196242262, 3.98511309186099783874256001543, 4.89518994682999238630700402437, 6.27839840305695842593853238188, 7.40738701223573808004123090406, 7.88212510254660794079181138333, 9.424647880610435644171278078648, 10.30796308701378352696202550950, 10.64040976016217645111400803766, 11.71344274319934092355729631513

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