Properties

Label 2-370-185.43-c1-0-1
Degree $2$
Conductor $370$
Sign $-0.997 + 0.0772i$
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 + (−2.18 + 2.18i)3-s − 4-s + (1.71 + 1.43i)5-s + (2.18 + 2.18i)6-s + (−3.43 + 3.43i)7-s + i·8-s − 6.54i·9-s + (1.43 − 1.71i)10-s − 3.68i·11-s + (2.18 − 2.18i)12-s − 1.41i·13-s + (3.43 + 3.43i)14-s + (−6.87 + 0.629i)15-s + 16-s − 4.10·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.26 + 1.26i)3-s − 0.5·4-s + (0.768 + 0.639i)5-s + (0.891 + 0.891i)6-s + (−1.29 + 1.29i)7-s + 0.353i·8-s − 2.18i·9-s + (0.452 − 0.543i)10-s − 1.10i·11-s + (0.630 − 0.630i)12-s − 0.391i·13-s + (0.918 + 0.918i)14-s + (−1.77 + 0.162i)15-s + 0.250·16-s − 0.996·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.997 + 0.0772i$
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: \((2,\ 370,\ (\ :1/2),\ -0.997 + 0.0772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00806507 - 0.208622i\)
\(L(\frac12)\) \(\approx\) \(0.00806507 - 0.208622i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 1.43i)T \)
37 \( 1 + (3.54 + 4.94i)T \)
good3 \( 1 + (2.18 - 2.18i)T - 3iT^{2} \)
7 \( 1 + (3.43 - 3.43i)T - 7iT^{2} \)
11 \( 1 + 3.68iT - 11T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
17 \( 1 + 4.10T + 17T^{2} \)
19 \( 1 + (1.16 - 1.16i)T - 19iT^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
29 \( 1 + (1.45 + 1.45i)T + 29iT^{2} \)
31 \( 1 + (6.49 - 6.49i)T - 31iT^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 - 6.72iT - 43T^{2} \)
47 \( 1 + (-1.97 + 1.97i)T - 47iT^{2} \)
53 \( 1 + (-2.68 - 2.68i)T + 53iT^{2} \)
59 \( 1 + (3.35 - 3.35i)T - 59iT^{2} \)
61 \( 1 + (5.14 - 5.14i)T - 61iT^{2} \)
67 \( 1 + (4.59 + 4.59i)T + 67iT^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 + (2.75 - 2.75i)T - 73iT^{2} \)
79 \( 1 + (-5.63 + 5.63i)T - 79iT^{2} \)
83 \( 1 + (1.45 + 1.45i)T + 83iT^{2} \)
89 \( 1 + (-7.40 - 7.40i)T + 89iT^{2} \)
97 \( 1 + 4.36T + 97T^{2} \)
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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

−11.64852466275371513107935977824, −10.67744142844087493169833158608, −10.42270901820036058648760661377, −9.259370764197386837534485431513, −8.941248762008040727816407532645, −6.46517526696266272816702406405, −5.96922821586205881327721604406, −5.15911021573426063951632492042, −3.66294778312106569969647965466, −2.70339034305590081894579535867, 0.15802028102805736043198459690, 1.75175067077963828433399166578, 4.24979060406134618691803002388, 5.36882671634449460895952656153, 6.34053724633025663923350085473, 6.95343909341544859605534506415, 7.53092429994376530384620945435, 9.137590455179072071641868713300, 9.983636518433535642938832429928, 10.92356124010965415144336802091

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