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Properties

Label 2-751-751.750-c0-0-7
Degree $2$
Conductor $751$
Sign $1$
Analytic cond. $0.374797$
Root an. cond. $0.612207$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.33·2-s + 0.790·4-s + 0.618·5-s − 0.279·8-s + 9-s + 0.827·10-s − 1.95·13-s − 1.16·16-s + 1.33·18-s − 19-s + 0.488·20-s + 1.82·23-s − 0.618·25-s − 2.61·26-s − 1.27·32-s + 0.790·36-s + 1.82·37-s − 1.33·38-s − 0.172·40-s − 1.61·43-s + 0.618·45-s + 2.44·46-s − 0.209·47-s + 49-s − 0.827·50-s − 1.54·52-s − 53-s + ⋯
L(s)  = 1  + 1.33·2-s + 0.790·4-s + 0.618·5-s − 0.279·8-s + 9-s + 0.827·10-s − 1.95·13-s − 1.16·16-s + 1.33·18-s − 19-s + 0.488·20-s + 1.82·23-s − 0.618·25-s − 2.61·26-s − 1.27·32-s + 0.790·36-s + 1.82·37-s − 1.33·38-s − 0.172·40-s − 1.61·43-s + 0.618·45-s + 2.44·46-s − 0.209·47-s + 49-s − 0.827·50-s − 1.54·52-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(751\)
Sign: $1$
Analytic conductor: \(0.374797\)
Root analytic conductor: \(0.612207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{751} (750, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 751,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863490454\)
\(L(\frac12)\) \(\approx\) \(1.863490454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad751 \( 1 - T \)
good2 \( 1 - 1.33T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.95T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - 1.82T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.82T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + 0.209T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + 1.95T + T^{2} \)
61 \( 1 + 0.209T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.82T + T^{2} \)
97 \( 1 - 1.33T + T^{2} \)
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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

−10.61223018657268273487427841724, −9.728891445429701924871204542715, −9.114309601179152673794809466991, −7.63635958496260471239748067920, −6.83455958065333368397658031563, −5.99071010174616003478566176049, −4.84539445927652408179862175217, −4.52518201471479982972334087685, −3.10011792275833001234299654176, −2.07934805041646926036070237994, 2.07934805041646926036070237994, 3.10011792275833001234299654176, 4.52518201471479982972334087685, 4.84539445927652408179862175217, 5.99071010174616003478566176049, 6.83455958065333368397658031563, 7.63635958496260471239748067920, 9.114309601179152673794809466991, 9.728891445429701924871204542715, 10.61223018657268273487427841724

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