Properties

Label 2-5415-1.1-c1-0-69
Degree $2$
Conductor $5415$
Sign $1$
Analytic cond. $43.2389$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 3-s − 1.82·4-s − 5-s − 0.414·6-s + 1.41·7-s − 1.58·8-s + 9-s − 0.414·10-s + 6.24·11-s + 1.82·12-s + 0.585·13-s + 0.585·14-s + 15-s + 3·16-s + 6.82·17-s + 0.414·18-s + 1.82·20-s − 1.41·21-s + 2.58·22-s − 3.65·23-s + 1.58·24-s + 25-s + 0.242·26-s − 27-s − 2.58·28-s + 1.41·29-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s − 0.169·6-s + 0.534·7-s − 0.560·8-s + 0.333·9-s − 0.130·10-s + 1.88·11-s + 0.527·12-s + 0.162·13-s + 0.156·14-s + 0.258·15-s + 0.750·16-s + 1.65·17-s + 0.0976·18-s + 0.408·20-s − 0.308·21-s + 0.551·22-s − 0.762·23-s + 0.323·24-s + 0.200·25-s + 0.0475·26-s − 0.192·27-s − 0.488·28-s + 0.262·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5415\)    =    \(3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(43.2389\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5415} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5415,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722830398\)
\(L(\frac12)\) \(\approx\) \(1.722830398\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - 0.414T + 2T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 8.82T + 31T^{2} \)
37 \( 1 - 0.585T + 37T^{2} \)
41 \( 1 + 8.24T + 41T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 - 3.65T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 1.65T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 - 3.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 7.17T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−8.146882334194692924228764148594, −7.57994782542911195428636110027, −6.45922099321992712793353631351, −6.07512096931034695897690850642, −5.11376750850109856156349982156, −4.52638310859673068871195850233, −3.84098886181288094226346267997, −3.22185462012852144922909021079, −1.52098484718172889798692620556, −0.78409358858171443815299128567, 0.78409358858171443815299128567, 1.52098484718172889798692620556, 3.22185462012852144922909021079, 3.84098886181288094226346267997, 4.52638310859673068871195850233, 5.11376750850109856156349982156, 6.07512096931034695897690850642, 6.45922099321992712793353631351, 7.57994782542911195428636110027, 8.146882334194692924228764148594

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