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Properties

Label 2-370-185.43-c1-0-17
Degree $2$
Conductor $370$
Sign $0.820 + 0.571i$
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  + i·2-s + (2.09 − 2.09i)3-s − 4-s + (1.30 − 1.81i)5-s + (2.09 + 2.09i)6-s + (−0.643 + 0.643i)7-s − i·8-s − 5.74i·9-s + (1.81 + 1.30i)10-s + 1.88i·11-s + (−2.09 + 2.09i)12-s − 0.536i·13-s + (−0.643 − 0.643i)14-s + (−1.06 − 6.52i)15-s + 16-s + 0.334·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (1.20 − 1.20i)3-s − 0.5·4-s + (0.583 − 0.812i)5-s + (0.853 + 0.853i)6-s + (−0.243 + 0.243i)7-s − 0.353i·8-s − 1.91i·9-s + (0.574 + 0.412i)10-s + 0.568i·11-s + (−0.603 + 0.603i)12-s − 0.148i·13-s + (−0.172 − 0.172i)14-s + (−0.276 − 1.68i)15-s + 0.250·16-s + 0.0811·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.820 + 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86462 - 0.585788i\)
\(L(\frac12)\) \(\approx\) \(1.86462 - 0.585788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-1.30 + 1.81i)T \)
37 \( 1 + (-6.02 - 0.819i)T \)
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 \( 1 + 0.536iT - 13T^{2} \)
17 \( 1 - 0.334T + 17T^{2} \)
19 \( 1 + (1.08 - 1.08i)T - 19iT^{2} \)
23 \( 1 - 4.65iT - 23T^{2} \)
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 \( 1 - 11.9iT - 43T^{2} \)
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 \( 1 + 17.1T + 97T^{2} \)
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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

−11.67152077915356406516092338900, −9.658907074438623036528476430451, −9.418583577238158065966638534288, −8.166560461793720641207739980549, −7.83903101980149919716201687789, −6.61826704166069095725479564662, −5.80603861818486443770503764236, −4.34680109757585745376955399990, −2.74293543952200716531778777994, −1.41149547565734070857009715608, 2.29900703654871576691779603968, 3.20715635934007995790402625976, 4.05198575313494048954641048967, 5.31652858539424527240717996225, 6.78804115222471572107297442856, 8.174368656119266383947202814368, 9.033155826084850217677001667884, 9.708208726087443661417091905334, 10.60043865431496791919488179957, 10.89730595894063972793628619963

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