Properties

Label 2-1300-1.1-c1-0-7
Degree $2$
Conductor $1300$
Sign $1$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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  − 0.539·3-s + 3.70·7-s − 2.70·9-s + 5.51·11-s + 13-s − 0.340·17-s − 2.24·19-s − 2·21-s + 1.46·23-s + 3.07·27-s + 0.630·29-s − 6.58·31-s − 2.97·33-s + 5.55·37-s − 0.539·39-s − 4.68·41-s + 10.6·43-s + 4.29·47-s + 6.75·49-s + 0.183·51-s + 9.07·53-s + 1.21·57-s + 0.986·59-s − 12.2·61-s − 10.0·63-s + 4.97·67-s − 0.787·69-s + ⋯
L(s)  = 1  − 0.311·3-s + 1.40·7-s − 0.903·9-s + 1.66·11-s + 0.277·13-s − 0.0825·17-s − 0.515·19-s − 0.436·21-s + 0.304·23-s + 0.592·27-s + 0.117·29-s − 1.18·31-s − 0.517·33-s + 0.912·37-s − 0.0863·39-s − 0.730·41-s + 1.62·43-s + 0.625·47-s + 0.965·49-s + 0.0256·51-s + 1.24·53-s + 0.160·57-s + 0.128·59-s − 1.56·61-s − 1.26·63-s + 0.607·67-s − 0.0948·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.805038455\)
\(L(\frac12)\) \(\approx\) \(1.805038455\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good3 \( 1 + 0.539T + 3T^{2} \)
7 \( 1 - 3.70T + 7T^{2} \)
11 \( 1 - 5.51T + 11T^{2} \)
17 \( 1 + 0.340T + 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 - 0.630T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
37 \( 1 - 5.55T + 37T^{2} \)
41 \( 1 + 4.68T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 4.29T + 47T^{2} \)
53 \( 1 - 9.07T + 53T^{2} \)
59 \( 1 - 0.986T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 4.97T + 67T^{2} \)
71 \( 1 - 7.32T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 9.75T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−9.439413409980495000851390196985, −8.818509721922320049439203965043, −8.162968339712225510090389357344, −7.18032987489071563681449185556, −6.25705376490648559717836476146, −5.49527405116416715578461019538, −4.53341978027223018104892456561, −3.70478772632439526469513490504, −2.23570288788591995743982877735, −1.08421902638743242855146018546, 1.08421902638743242855146018546, 2.23570288788591995743982877735, 3.70478772632439526469513490504, 4.53341978027223018104892456561, 5.49527405116416715578461019538, 6.25705376490648559717836476146, 7.18032987489071563681449185556, 8.162968339712225510090389357344, 8.818509721922320049439203965043, 9.439413409980495000851390196985

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