Properties

Label 2-348726-1.1-c1-0-20
Degree $2$
Conductor $348726$
Sign $-1$
Analytic cond. $2784.59$
Root an. cond. $52.7692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s + 7-s − 8-s + 9-s + 3·10-s + 3·11-s + 12-s − 4·13-s − 14-s − 3·15-s + 16-s + 3·17-s − 18-s − 3·20-s + 21-s − 3·22-s − 23-s − 24-s + 4·25-s + 4·26-s + 27-s + 28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.670·20-s + 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348726\)    =    \(2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(2784.59\)
Root analytic conductor: \(52.7692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ -1)\)

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

−12.53421390324032, −12.29458672718143, −11.74617004551065, −11.45223437851182, −11.11557492559205, −10.27626231713582, −10.17170280040195, −9.336936799932691, −9.254670417847986, −8.549796582186892, −8.260348375817296, −7.598851307084920, −7.504796938669423, −7.054952750254601, −6.510454816625842, −5.815329633314113, −5.211771914109769, −4.672493448432010, −4.127817538638091, −3.584742292458917, −3.292162410660212, −2.612885681212402, −1.802599715087442, −1.552696001728066, −0.6154242802410732, 0, 0.6154242802410732, 1.552696001728066, 1.802599715087442, 2.612885681212402, 3.292162410660212, 3.584742292458917, 4.127817538638091, 4.672493448432010, 5.211771914109769, 5.815329633314113, 6.510454816625842, 7.054952750254601, 7.504796938669423, 7.598851307084920, 8.260348375817296, 8.549796582186892, 9.254670417847986, 9.336936799932691, 10.17170280040195, 10.27626231713582, 11.11557492559205, 11.45223437851182, 11.74617004551065, 12.29458672718143, 12.53421390324032

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