Properties

Label 2-19110-1.1-c1-0-39
Degree $2$
Conductor $19110$
Sign $1$
Analytic cond. $152.594$
Root an. cond. $12.3528$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s + 15-s + 16-s + 6·17-s + 18-s + 20-s + 4·22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s − 6·29-s + 30-s − 8·31-s + 32-s + 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.223·20-s + 0.852·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19110\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(152.594\)
Root analytic conductor: \(12.3528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 19110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.251321193\)
\(L(\frac12)\) \(\approx\) \(6.251321193\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.65202308376742, −14.85264016839545, −14.52830887079703, −14.19406204589258, −13.70970479188483, −12.86621748813642, −12.67452876377118, −11.96320121102064, −11.36013785611006, −10.87406757367272, −9.987709937658778, −9.592020751372101, −9.113907290451449, −8.294228873814501, −7.664256638700322, −7.201206233264082, −6.351288861011883, −5.848748842502357, −5.380018385506017, −4.337567943966967, −3.833751417777536, −3.360507437618922, −2.398857018236452, −1.748850176493311, −0.9534775064249759, 0.9534775064249759, 1.748850176493311, 2.398857018236452, 3.360507437618922, 3.833751417777536, 4.337567943966967, 5.380018385506017, 5.848748842502357, 6.351288861011883, 7.201206233264082, 7.664256638700322, 8.294228873814501, 9.113907290451449, 9.592020751372101, 9.987709937658778, 10.87406757367272, 11.36013785611006, 11.96320121102064, 12.67452876377118, 12.86621748813642, 13.70970479188483, 14.19406204589258, 14.52830887079703, 14.85264016839545, 15.65202308376742

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