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Properties

Label 2-370-185.104-c1-0-6
Degree $2$
Conductor $370$
Sign $0.195 - 0.980i$
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  + (0.939 − 0.342i)2-s + (−1.00 + 2.77i)3-s + (0.766 − 0.642i)4-s + (1.26 + 1.84i)5-s + 2.94i·6-s + (3.81 − 0.672i)7-s + (0.500 − 0.866i)8-s + (−4.35 − 3.65i)9-s + (1.81 + 1.30i)10-s + (0.398 − 0.690i)11-s + (1.00 + 2.77i)12-s + (−2.06 + 1.73i)13-s + (3.35 − 1.93i)14-s + (−6.38 + 1.64i)15-s + (0.173 − 0.984i)16-s + (−0.125 − 0.105i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.582 + 1.59i)3-s + (0.383 − 0.321i)4-s + (0.565 + 0.824i)5-s + 1.20i·6-s + (1.44 − 0.254i)7-s + (0.176 − 0.306i)8-s + (−1.45 − 1.21i)9-s + (0.575 + 0.411i)10-s + (0.120 − 0.208i)11-s + (0.291 + 0.799i)12-s + (−0.573 + 0.480i)13-s + (0.896 − 0.517i)14-s + (−1.64 + 0.424i)15-s + (0.0434 − 0.246i)16-s + (−0.0303 − 0.0254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49157 + 1.22418i\)
\(L(\frac12)\) \(\approx\) \(1.49157 + 1.22418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-1.26 - 1.84i)T \)
37 \( 1 + (-2.54 - 5.52i)T \)
good3 \( 1 + (1.00 - 2.77i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-3.81 + 0.672i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.398 + 0.690i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.06 - 1.73i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.125 + 0.105i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.41 - 3.87i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (3.01 + 5.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.297 - 0.171i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.59iT - 31T^{2} \)
41 \( 1 + (-8.50 + 7.13i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 9.32T + 43T^{2} \)
47 \( 1 + (-6.52 + 3.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.98 - 1.23i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.82 + 0.851i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.93 + 7.07i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (8.06 - 1.42i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.42 - 1.97i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 7.77iT - 73T^{2} \)
79 \( 1 + (-9.00 + 1.58i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-5.14 + 6.13i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-6.57 - 1.15i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-7.58 - 13.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.43421490337019989090094914641, −10.64058312395547940065338055502, −10.22269353854305057215743991149, −9.214149998003472302946921451195, −7.84467517805482644386608801072, −6.38481729422886802355833000581, −5.51144493876258448718140731743, −4.57472268328794257616943507170, −3.84312917081585312881335266140, −2.25320716234208862547298950010, 1.34634878591018160528666801597, 2.34028286455147006857295255330, 4.70650244449840901976753933357, 5.38890820375675745384868156487, 6.25075309064370917972977207635, 7.42482560585605432748063984491, 8.000713406621502134293029977732, 9.036735290293254444997635541519, 10.69421321583667142273451829624, 11.68036350766616807167788697833

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