Properties

Label 2-7406-1.1-c1-0-233
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s − 2·12-s − 4·13-s + 14-s + 16-s − 6·17-s − 18-s − 2·19-s + 2·21-s + 2·24-s − 5·25-s + 4·26-s + 4·27-s − 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s − 2·37-s + 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-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: \(2\)
Selberg data: \((2,\ 7406,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.13744962889432520512169772456, −6.48201705093403727879604698896, −6.02025859781944321398556535937, −5.15792369029560896264352021468, −4.55285159668134168351693053653, −3.51499835308930547509430841574, −2.44758310999345700749523202055, −1.66340644056464459884005604614, 0, 0, 1.66340644056464459884005604614, 2.44758310999345700749523202055, 3.51499835308930547509430841574, 4.55285159668134168351693053653, 5.15792369029560896264352021468, 6.02025859781944321398556535937, 6.48201705093403727879604698896, 7.13744962889432520512169772456

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