Properties

Label 2-350658-1.1-c1-0-33
Degree $2$
Conductor $350658$
Sign $1$
Analytic cond. $2800.01$
Root an. cond. $52.9151$
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 − 7-s − 8-s + 14-s + 16-s + 6·17-s + 6·19-s + 23-s − 5·25-s − 28-s + 10·29-s + 4·31-s − 32-s − 6·34-s − 2·37-s − 6·38-s − 10·41-s + 4·43-s − 46-s − 12·47-s + 49-s + 5·50-s + 6·53-s + 56-s − 10·58-s + 2·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.208·23-s − 25-s − 0.188·28-s + 1.85·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.973·38-s − 1.56·41-s + 0.609·43-s − 0.147·46-s − 1.75·47-s + 1/7·49-s + 0.707·50-s + 0.824·53-s + 0.133·56-s − 1.31·58-s + 0.260·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350658\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(2800.01\)
Root analytic conductor: \(52.9151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{350658} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 350658,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033535258\)
\(L(\frac12)\) \(\approx\) \(2.033535258\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.33294110720839, −11.95192113201305, −11.77468301160906, −11.21706755714354, −10.58268493847131, −10.02115981148222, −9.898097482860008, −9.594790081855788, −8.809548653315892, −8.463592817469916, −7.939240753901053, −7.616826967012398, −7.062980602939653, −6.549643594872910, −6.175924140322597, −5.535719258211038, −5.099507026757145, −4.653843495171980, −3.639001749501182, −3.479164254793211, −2.867954741048379, −2.340815147688335, −1.510799995198933, −1.058179605200749, −0.4768277442762474, 0.4768277442762474, 1.058179605200749, 1.510799995198933, 2.340815147688335, 2.867954741048379, 3.479164254793211, 3.639001749501182, 4.653843495171980, 5.099507026757145, 5.535719258211038, 6.175924140322597, 6.549643594872910, 7.062980602939653, 7.616826967012398, 7.939240753901053, 8.463592817469916, 8.809548653315892, 9.594790081855788, 9.898097482860008, 10.02115981148222, 10.58268493847131, 11.21706755714354, 11.77468301160906, 11.95192113201305, 12.33294110720839

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