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Properties

Label 2-371-371.153-c0-0-0
Degree $2$
Conductor $371$
Sign $0.899 - 0.437i$
Analytic cond. $0.185153$
Root an. cond. $0.430294$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.213 − 0.112i)2-s + (−0.535 + 0.775i)4-s + (0.885 − 0.464i)7-s + (−0.0564 + 0.464i)8-s + (−0.748 + 0.663i)9-s + (1.45 + 0.358i)11-s + (0.136 − 0.198i)14-s + (−0.293 − 0.775i)16-s + (−0.0854 + 0.225i)18-s + (0.350 − 0.0863i)22-s − 1.94·23-s + (0.120 − 0.992i)25-s + (−0.113 + 0.935i)28-s + (−0.234 + 0.0576i)29-s + (−0.499 − 0.442i)32-s + ⋯
L(s)  = 1  + (0.213 − 0.112i)2-s + (−0.535 + 0.775i)4-s + (0.885 − 0.464i)7-s + (−0.0564 + 0.464i)8-s + (−0.748 + 0.663i)9-s + (1.45 + 0.358i)11-s + (0.136 − 0.198i)14-s + (−0.293 − 0.775i)16-s + (−0.0854 + 0.225i)18-s + (0.350 − 0.0863i)22-s − 1.94·23-s + (0.120 − 0.992i)25-s + (−0.113 + 0.935i)28-s + (−0.234 + 0.0576i)29-s + (−0.499 − 0.442i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(0.185153\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 371,\ (\ :0),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8656873538\)
\(L(\frac12)\) \(\approx\) \(0.8656873538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.885 + 0.464i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
good2 \( 1 + (-0.213 + 0.112i)T + (0.568 - 0.822i)T^{2} \)
3 \( 1 + (0.748 - 0.663i)T^{2} \)
5 \( 1 + (-0.120 + 0.992i)T^{2} \)
11 \( 1 + (-1.45 - 0.358i)T + (0.885 + 0.464i)T^{2} \)
13 \( 1 + (0.354 - 0.935i)T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + (0.354 - 0.935i)T^{2} \)
23 \( 1 + 1.94T + T^{2} \)
29 \( 1 + (0.234 - 0.0576i)T + (0.885 - 0.464i)T^{2} \)
31 \( 1 + (-0.885 + 0.464i)T^{2} \)
37 \( 1 + (0.402 + 1.06i)T + (-0.748 + 0.663i)T^{2} \)
41 \( 1 + (-0.885 - 0.464i)T^{2} \)
43 \( 1 + (-0.251 + 0.663i)T + (-0.748 - 0.663i)T^{2} \)
47 \( 1 + (-0.120 - 0.992i)T^{2} \)
59 \( 1 + (-0.120 - 0.992i)T^{2} \)
61 \( 1 + (0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.850 - 1.23i)T + (-0.354 - 0.935i)T^{2} \)
71 \( 1 + (0.627 - 1.65i)T + (-0.748 - 0.663i)T^{2} \)
73 \( 1 + (0.970 - 0.239i)T^{2} \)
79 \( 1 + (-1.56 - 0.822i)T + (0.568 + 0.822i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.970 - 0.239i)T^{2} \)
97 \( 1 + (-0.120 + 0.992i)T^{2} \)
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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

−11.81015281230494833936685785642, −10.98696246737174283664749550545, −9.824742785780269567233034414277, −8.691482173651182845384653380426, −8.108853812911274267024369283008, −7.13318723994552197275446125801, −5.73576620061417404113450346163, −4.51264416854111989436333682269, −3.81236753341625676184926495464, −2.13470113557704686708159660004, 1.57342197134840820375850009350, 3.58178331069953968189051650545, 4.68133898259840049394501140039, 5.84483710563603606394867590722, 6.39369200569590341331695174284, 7.983726509654205905685599625255, 8.981208973318923440851421555035, 9.463891174374996103726898608449, 10.72234718208317184783476869282, 11.66467527492826692761107742442

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