Properties

Label 2-7406-1.1-c1-0-105
Degree $2$
Conductor $7406$
Sign $1$
Analytic cond. $59.1372$
Root an. cond. $7.69007$
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 + 4-s + 2·5-s − 7-s − 8-s − 3·9-s − 2·10-s + 4·11-s + 4·13-s + 14-s + 16-s + 8·17-s + 3·18-s + 2·19-s + 2·20-s − 4·22-s − 25-s − 4·26-s − 28-s + 2·29-s − 6·31-s − 32-s − 8·34-s − 2·35-s − 3·36-s + 10·37-s − 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.632·10-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.707·18-s + 0.458·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.371·29-s − 1.07·31-s − 0.176·32-s − 1.37·34-s − 0.338·35-s − 1/2·36-s + 1.64·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7406\)    =    \(2 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(59.1372\)
Root analytic conductor: \(7.69007\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7406} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7406,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.027638188\)
\(L(\frac12)\) \(\approx\) \(2.027638188\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−7.88344761387658947503598101916, −7.37784559335912029840696264493, −6.28664941237184942144060567566, −5.92214710684541833965840069767, −5.56528813677650326141936298733, −4.15496858462953392394573093342, −3.34561366065603947604442305947, −2.67929126530051147823213321473, −1.52536654790130646687896240029, −0.877236400762477963039132426390, 0.877236400762477963039132426390, 1.52536654790130646687896240029, 2.67929126530051147823213321473, 3.34561366065603947604442305947, 4.15496858462953392394573093342, 5.56528813677650326141936298733, 5.92214710684541833965840069767, 6.28664941237184942144060567566, 7.37784559335912029840696264493, 7.88344761387658947503598101916

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