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Properties

Label 2-370-185.117-c1-0-8
Degree $2$
Conductor $370$
Sign $0.575 - 0.817i$
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 + (1 + i)3-s + 4-s + (−1 + 2i)5-s + (1 + i)6-s + (1 + i)7-s + 8-s − i·9-s + (−1 + 2i)10-s − 2i·11-s + (1 + i)12-s + 2·13-s + (1 + i)14-s + (−3 + i)15-s + 16-s + 4i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.577 + 0.577i)3-s + 0.5·4-s + (−0.447 + 0.894i)5-s + (0.408 + 0.408i)6-s + (0.377 + 0.377i)7-s + 0.353·8-s − 0.333i·9-s + (−0.316 + 0.632i)10-s − 0.603i·11-s + (0.288 + 0.288i)12-s + 0.554·13-s + (0.267 + 0.267i)14-s + (−0.774 + 0.258i)15-s + 0.250·16-s + 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.575 - 0.817i$
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: \((2,\ 370,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05922 + 1.06930i\)
\(L(\frac12)\) \(\approx\) \(2.05922 + 1.06930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
37 \( 1 + (1 + 6i)T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + (5 - 5i)T - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 + (-7 + 7i)T - 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (7 + 7i)T + 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (1 - i)T - 59iT^{2} \)
61 \( 1 + (3 - 3i)T - 61iT^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-9 - 9i)T + 73iT^{2} \)
79 \( 1 + (1 - i)T - 79iT^{2} \)
83 \( 1 + (5 - 5i)T - 83iT^{2} \)
89 \( 1 + (-5 - 5i)T + 89iT^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−11.51681049479976715515495458736, −10.72214498184088405583760294808, −9.913561959008795010014553711726, −8.550678622946176948223109707713, −7.968092296445966825260904727649, −6.49397020060739909292475709936, −5.84108691251726407248319257508, −4.10728915493454226033796375498, −3.64159708286768808345882270281, −2.32661717346622627370675610005, 1.48399474842316005654978594715, 2.91408694033091515323513245375, 4.46225439661671786023374444253, 4.99367285908152138982104726730, 6.61342180482473715016656383816, 7.49965213761600358516025530418, 8.309662155208741088838408339798, 9.173798326193050005880679000962, 10.60314747515121830693942006683, 11.43854637309066998618799233508

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