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Properties

Label 2-370-185.104-c1-0-8
Degree $2$
Conductor $370$
Sign $0.965 + 0.261i$
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 + (0.362 − 0.995i)3-s + (0.766 − 0.642i)4-s + (2.23 − 0.0855i)5-s + 1.05i·6-s + (1.06 − 0.187i)7-s + (−0.500 + 0.866i)8-s + (1.43 + 1.20i)9-s + (−2.07 + 0.844i)10-s + (0.436 − 0.755i)11-s + (−0.362 − 0.995i)12-s + (−1.18 + 0.992i)13-s + (−0.933 + 0.539i)14-s + (0.724 − 2.25i)15-s + (0.173 − 0.984i)16-s + (0.626 + 0.525i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.209 − 0.574i)3-s + (0.383 − 0.321i)4-s + (0.999 − 0.0382i)5-s + 0.432i·6-s + (0.401 − 0.0707i)7-s + (−0.176 + 0.306i)8-s + (0.479 + 0.402i)9-s + (−0.654 + 0.267i)10-s + (0.131 − 0.227i)11-s + (−0.104 − 0.287i)12-s + (−0.328 + 0.275i)13-s + (−0.249 + 0.144i)14-s + (0.187 − 0.582i)15-s + (0.0434 − 0.246i)16-s + (0.151 + 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.965 + 0.261i$
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: \((2,\ 370,\ (\ :1/2),\ 0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33178 - 0.177105i\)
\(L(\frac12)\) \(\approx\) \(1.33178 - 0.177105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.0855i)T \)
37 \( 1 + (-4.86 - 3.64i)T \)
good3 \( 1 + (-0.362 + 0.995i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.06 + 0.187i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.436 + 0.755i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.18 - 0.992i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.626 - 0.525i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.570 - 1.56i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (2.32 + 4.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.84 + 1.06i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.18iT - 31T^{2} \)
41 \( 1 + (-1.27 + 1.06i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 2.12T + 43T^{2} \)
47 \( 1 + (-4.23 + 2.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.91 + 0.865i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (10.8 + 1.90i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.22 - 1.45i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.75 + 1.01i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (10.2 + 3.71i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + (4.86 - 0.858i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.61 - 3.11i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (15.0 + 2.66i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (5.51 + 9.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.15126258888835201000305632541, −10.25885685708352099856142695519, −9.545298222973782372099519868704, −8.485587321230322976226930905208, −7.69456714689575516082007119843, −6.69911464020477873475385344005, −5.82824060953592961623872288623, −4.54403692759087823397713800046, −2.47924273863882558861653095602, −1.44263681158732501516862899886, 1.56126918573315380508468459800, 2.96073790367735540272843749188, 4.39023360288806083654100335434, 5.62011239370022229185555712017, 6.79719285681157756690442463415, 7.82155385870382322009994755301, 9.080643334686886288764874349364, 9.546201179054322207807334668853, 10.31978871328523238831526329590, 11.13905342667411000347012444173

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