Properties

Label 2-370-185.43-c1-0-8
Degree $2$
Conductor $370$
Sign $-0.101 - 0.994i$
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 + (−0.536 + 0.536i)3-s − 4-s + (−0.127 − 2.23i)5-s + (−0.536 − 0.536i)6-s + (0.767 − 0.767i)7-s i·8-s + 2.42i·9-s + (2.23 − 0.127i)10-s + 4.39i·11-s + (0.536 − 0.536i)12-s + 6.74i·13-s + (0.767 + 0.767i)14-s + (1.26 + 1.12i)15-s + 16-s + 7.34·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.309 + 0.309i)3-s − 0.5·4-s + (−0.0571 − 0.998i)5-s + (−0.219 − 0.219i)6-s + (0.290 − 0.290i)7-s − 0.353i·8-s + 0.808i·9-s + (0.705 − 0.0404i)10-s + 1.32i·11-s + (0.154 − 0.154i)12-s + 1.87i·13-s + (0.205 + 0.205i)14-s + (0.326 + 0.291i)15-s + 0.250·16-s + 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.101 - 0.994i$
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: \((2,\ 370,\ (\ :1/2),\ -0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770237 + 0.853236i\)
\(L(\frac12)\) \(\approx\) \(0.770237 + 0.853236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.127 + 2.23i)T \)
37 \( 1 + (-0.633 + 6.04i)T \)
good3 \( 1 + (0.536 - 0.536i)T - 3iT^{2} \)
7 \( 1 + (-0.767 + 0.767i)T - 7iT^{2} \)
11 \( 1 - 4.39iT - 11T^{2} \)
13 \( 1 - 6.74iT - 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + (-2.59 + 2.59i)T - 19iT^{2} \)
23 \( 1 - 1.20iT - 23T^{2} \)
29 \( 1 + (-1.25 - 1.25i)T + 29iT^{2} \)
31 \( 1 + (4.14 - 4.14i)T - 31iT^{2} \)
41 \( 1 - 4.07iT - 41T^{2} \)
43 \( 1 + 8.56iT - 43T^{2} \)
47 \( 1 + (-7.68 + 7.68i)T - 47iT^{2} \)
53 \( 1 + (-2.31 - 2.31i)T + 53iT^{2} \)
59 \( 1 + (7.61 - 7.61i)T - 59iT^{2} \)
61 \( 1 + (-1.14 + 1.14i)T - 61iT^{2} \)
67 \( 1 + (6.25 + 6.25i)T + 67iT^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (-1.88 + 1.88i)T - 73iT^{2} \)
79 \( 1 + (-5.13 + 5.13i)T - 79iT^{2} \)
83 \( 1 + (-0.570 - 0.570i)T + 83iT^{2} \)
89 \( 1 + (-7.54 - 7.54i)T + 89iT^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−11.87775403334374526129154984534, −10.58538539327553507647788555620, −9.586734391055184843640757941294, −8.962013743652708927951473987377, −7.67551358557302611567445729102, −7.15206200398025890514915211627, −5.58868235098394332258762137875, −4.84058815952617061492181405325, −4.09530760749557039653801238308, −1.66411611642409934407879915869, 0.912822125550698145406151575044, 2.99610960525904060551216143003, 3.51313607123125065790536073523, 5.57743126617216937115142662559, 6.00781506788362678942224695605, 7.58969780413790904956695628062, 8.245143365650925993268842686868, 9.639765280023336698551627948769, 10.38841945395520523691190592148, 11.20105363658333859369729004895

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