Properties

Label 2-10626-1.1-c1-0-2
Degree $2$
Conductor $10626$
Sign $1$
Analytic cond. $84.8490$
Root an. cond. $9.21135$
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 − 6-s − 7-s + 8-s + 9-s + 11-s − 12-s − 13-s − 14-s + 16-s + 5·17-s + 18-s − 5·19-s + 21-s + 22-s + 23-s − 24-s − 5·25-s − 26-s − 27-s − 28-s + 9·29-s − 5·31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 1.14·19-s + 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.898·31-s + 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10626\)    =    \(2 \cdot 3 \cdot 7 \cdot 11 \cdot 23\)
Sign: $1$
Analytic conductor: \(84.8490\)
Root analytic conductor: \(9.21135\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10626} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10626,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.500847088\)
\(L(\frac12)\) \(\approx\) \(2.500847088\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
23 \( 1 - T \)
good5 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.41855390735566, −16.05046023283788, −15.41139040375707, −14.78702332269516, −14.23798994704435, −13.75677436890663, −12.96883916651161, −12.35942444107372, −12.23677276780986, −11.41248104281667, −10.78724342960705, −10.27513648744374, −9.650479265740951, −8.938921642023874, −8.051392384464445, −7.459057298522546, −6.735621861962972, −6.152839276373167, −5.634693236927321, −4.889615581100854, −4.182440524760405, −3.547914241350204, −2.674843850277720, −1.774131681636237, −0.6826963345723457, 0.6826963345723457, 1.774131681636237, 2.674843850277720, 3.547914241350204, 4.182440524760405, 4.889615581100854, 5.634693236927321, 6.152839276373167, 6.735621861962972, 7.459057298522546, 8.051392384464445, 8.938921642023874, 9.650479265740951, 10.27513648744374, 10.78724342960705, 11.41248104281667, 12.23677276780986, 12.35942444107372, 12.96883916651161, 13.75677436890663, 14.23798994704435, 14.78702332269516, 15.41139040375707, 16.05046023283788, 16.41855390735566

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