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Properties

Label 2-370-185.82-c1-0-14
Degree $2$
Conductor $370$
Sign $0.571 + 0.820i$
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  + (−0.5 − 0.866i)2-s + (2.36 − 0.633i)3-s + (−0.499 + 0.866i)4-s + (2.23 + 0.133i)5-s + (−1.73 − 1.73i)6-s + (−1.36 + 0.366i)7-s + 0.999·8-s + (2.59 − 1.50i)9-s + (−1 − 1.99i)10-s + 0.464i·11-s + (−0.633 + 2.36i)12-s + (1.86 − 3.23i)13-s + (1 + 0.999i)14-s + (5.36 − 1.09i)15-s + (−0.5 − 0.866i)16-s + (−0.633 + 0.366i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (1.36 − 0.366i)3-s + (−0.249 + 0.433i)4-s + (0.998 + 0.0599i)5-s + (−0.707 − 0.707i)6-s + (−0.516 + 0.138i)7-s + 0.353·8-s + (0.866 − 0.500i)9-s + (−0.316 − 0.632i)10-s + 0.139i·11-s + (−0.183 + 0.683i)12-s + (0.517 − 0.896i)13-s + (0.267 + 0.267i)14-s + (1.38 − 0.283i)15-s + (−0.125 − 0.216i)16-s + (−0.153 + 0.0887i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.571 + 0.820i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.571 + 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66002 - 0.866541i\)
\(L(\frac12)\) \(\approx\) \(1.66002 - 0.866541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-2.23 - 0.133i)T \)
37 \( 1 + (-0.5 + 6.06i)T \)
good3 \( 1 + (-2.36 + 0.633i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.36 - 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 - 0.464iT - 11T^{2} \)
13 \( 1 + (-1.86 + 3.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.633 - 0.366i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.133 - 0.5i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.267T + 23T^{2} \)
29 \( 1 + (-1.46 - 1.46i)T + 29iT^{2} \)
31 \( 1 + (6.19 - 6.19i)T - 31iT^{2} \)
41 \( 1 + (8.19 + 4.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2.09 - 2.09i)T + 47iT^{2} \)
53 \( 1 + (8.56 + 2.29i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.59 + 1.23i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.562 + 2.09i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.83 - 10.5i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.92 - 5.92i)T + 73iT^{2} \)
79 \( 1 + (0.928 + 3.46i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (5.46 + 1.46i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (3.79 - 14.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 9.46iT - 97T^{2} \)
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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

−10.98184040285676147418571133381, −10.18383983678216496653769048922, −9.288582451189711716396852231721, −8.747530132366224493420551960631, −7.78566148444406045363332804363, −6.69266371141125155498107924010, −5.38905436381975429039984707839, −3.58898025443676707729640172775, −2.74754758788002533090066621728, −1.64484797522181112494170960862, 1.90430715497974095113563083030, 3.26037347577718321093170999495, 4.54478443248201076265036428234, 5.96965622563691283398462359847, 6.82877862008383579159926181741, 8.030590904142064870295466686322, 8.901180083656096782698179712633, 9.479002106371631254629558278326, 10.08809751202857503848918858172, 11.29028508372073828229005911945

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