Properties

Label 2-348726-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s + 3·11-s − 12-s + 6·13-s + 14-s − 2·15-s + 16-s + 6·17-s − 18-s + 2·20-s + 21-s − 3·22-s + 23-s + 24-s − 25-s − 6·26-s − 27-s − 28-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.365·30-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: \(0\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.639167401\)
\(L(\frac12)\) \(\approx\) \(2.639167401\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−12.43858507183493, −11.91882877078566, −11.62478718825175, −11.24108407603428, −10.55381583992897, −10.25744932011241, −9.790276246329706, −9.574288953252393, −8.897731171148926, −8.508793512628891, −8.079134910453932, −7.511177314337070, −6.786365337807233, −6.506181425253967, −6.045432840003630, −5.865008031179678, −5.081593908244060, −4.698419028007575, −3.756674481631849, −3.397015473261235, −3.029547793566813, −1.917827096079351, −1.655891719036653, −1.072319508156785, −0.5579481328107797, 0.5579481328107797, 1.072319508156785, 1.655891719036653, 1.917827096079351, 3.029547793566813, 3.397015473261235, 3.756674481631849, 4.698419028007575, 5.081593908244060, 5.865008031179678, 6.045432840003630, 6.506181425253967, 6.786365337807233, 7.511177314337070, 8.079134910453932, 8.508793512628891, 8.897731171148926, 9.574288953252393, 9.790276246329706, 10.25744932011241, 10.55381583992897, 11.24108407603428, 11.62478718825175, 11.91882877078566, 12.43858507183493

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