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Properties

Label 2-370-185.117-c1-0-3
Degree $2$
Conductor $370$
Sign $0.960 - 0.278i$
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 + (−0.794 − 0.794i)3-s + 4-s + (2.22 + 0.217i)5-s + (0.794 + 0.794i)6-s + (2.93 + 2.93i)7-s − 8-s − 1.73i·9-s + (−2.22 − 0.217i)10-s + 3.55i·11-s + (−0.794 − 0.794i)12-s − 4.41·13-s + (−2.93 − 2.93i)14-s + (−1.59 − 1.94i)15-s + 16-s + 5.37i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.458 − 0.458i)3-s + 0.5·4-s + (0.995 + 0.0971i)5-s + (0.324 + 0.324i)6-s + (1.11 + 1.11i)7-s − 0.353·8-s − 0.578i·9-s + (−0.703 − 0.0686i)10-s + 1.07i·11-s + (−0.229 − 0.229i)12-s − 1.22·13-s + (−0.785 − 0.785i)14-s + (−0.412 − 0.501i)15-s + 0.250·16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.960 - 0.278i$
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: \((2,\ 370,\ (\ :1/2),\ 0.960 - 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07203 + 0.152151i\)
\(L(\frac12)\) \(\approx\) \(1.07203 + 0.152151i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.217i)T \)
37 \( 1 + (3.88 + 4.67i)T \)
good3 \( 1 + (0.794 + 0.794i)T + 3iT^{2} \)
7 \( 1 + (-2.93 - 2.93i)T + 7iT^{2} \)
11 \( 1 - 3.55iT - 11T^{2} \)
13 \( 1 + 4.41T + 13T^{2} \)
17 \( 1 - 5.37iT - 17T^{2} \)
19 \( 1 + (-1.98 + 1.98i)T - 19iT^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 + (-2.18 - 2.18i)T + 29iT^{2} \)
31 \( 1 + (-5.42 + 5.42i)T - 31iT^{2} \)
41 \( 1 + 6.34iT - 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + (-1.82 - 1.82i)T + 47iT^{2} \)
53 \( 1 + (9.57 - 9.57i)T - 53iT^{2} \)
59 \( 1 + (-4.44 + 4.44i)T - 59iT^{2} \)
61 \( 1 + (-1.44 + 1.44i)T - 61iT^{2} \)
67 \( 1 + (-7.88 + 7.88i)T - 67iT^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + (4.02 + 4.02i)T + 73iT^{2} \)
79 \( 1 + (-2.03 + 2.03i)T - 79iT^{2} \)
83 \( 1 + (4.51 - 4.51i)T - 83iT^{2} \)
89 \( 1 + (-2.13 - 2.13i)T + 89iT^{2} \)
97 \( 1 + 5.91iT - 97T^{2} \)
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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

−11.42964670132454060702834745774, −10.46776951170922528118666944345, −9.470754951881793384283145919485, −8.881845044047381395547531017892, −7.63950330732654339633570755720, −6.74714326627825856826900219914, −5.74449041787422537453894757237, −4.86125689953495117845036051287, −2.52880547176859144573578115747, −1.56838593941602592229876276462, 1.13243733653321496779901018892, 2.76775290076650559342957943992, 4.77753966138289939247282244965, 5.28909296845585937868484866205, 6.74805779448185809762994351184, 7.67071331267999864914369787297, 8.617667894641725297588222596649, 9.827548396598863810634408560008, 10.30497663616130942698089073243, 11.15977843870280306329852098597

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