Properties

Label 2-10626-1.1-c1-0-7
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 + 6·13-s + 14-s + 16-s + 4·17-s + 18-s − 2·19-s + 21-s − 22-s − 23-s + 24-s − 5·25-s + 6·26-s + 27-s + 28-s + 2·29-s − 2·31-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 + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.218·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.359·31-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

\(L(1)\) \(\approx\) \(5.213186679\)
\(L(\frac12)\) \(\approx\) \(5.213186679\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.31060872981060, −15.85740748109503, −15.37782318191036, −14.77071149752870, −14.15116449431198, −13.79341812887435, −13.16145220026471, −12.73671844690244, −11.92543740015749, −11.46343995147978, −10.70580963495858, −10.31135866395374, −9.465991006962730, −8.777542521065535, −8.068814616246664, −7.766876234124967, −6.859567447264466, −6.120418174010684, −5.621935998285525, −4.825759041035416, −3.878332217802725, −3.650505223973511, −2.657486479039091, −1.860563469187487, −0.9824204068290060, 0.9824204068290060, 1.860563469187487, 2.657486479039091, 3.650505223973511, 3.878332217802725, 4.825759041035416, 5.621935998285525, 6.120418174010684, 6.859567447264466, 7.766876234124967, 8.068814616246664, 8.777542521065535, 9.465991006962730, 10.31135866395374, 10.70580963495858, 11.46343995147978, 11.92543740015749, 12.73671844690244, 13.16145220026471, 13.79341812887435, 14.15116449431198, 14.77071149752870, 15.37782318191036, 15.85740748109503, 16.31060872981060

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