Properties

Label 2-348726-1.1-c1-0-11
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 + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 5·11-s + 12-s − 4·13-s + 14-s + 15-s + 16-s + 3·17-s − 18-s + 20-s − 21-s − 5·22-s + 23-s − 24-s − 4·25-s + 4·26-s + 27-s − 28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s − 1.06·22-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 + 0.371·29-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: $\chi_{348726} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.691166685\)
\(L(\frac12)\) \(\approx\) \(2.691166685\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.47771419321383, −12.05531545653893, −11.71354868599947, −11.21136867378348, −10.54710274874046, −9.972786614815695, −9.835280655809878, −9.430255651564193, −9.013515960848288, −8.438192850066345, −8.099068082818274, −7.511129346889008, −6.993940156823901, −6.644003309442194, −6.270697572510858, −5.529079333977321, −5.147791165896544, −4.363658163896473, −3.930281574833951, −3.246020194922113, −2.941602269745740, −2.120174338896798, −1.800558947641510, −1.122771897919575, −0.4841013593395477, 0.4841013593395477, 1.122771897919575, 1.800558947641510, 2.120174338896798, 2.941602269745740, 3.246020194922113, 3.930281574833951, 4.363658163896473, 5.147791165896544, 5.529079333977321, 6.270697572510858, 6.644003309442194, 6.993940156823901, 7.511129346889008, 8.099068082818274, 8.438192850066345, 9.013515960848288, 9.430255651564193, 9.835280655809878, 9.972786614815695, 10.54710274874046, 11.21136867378348, 11.71354868599947, 12.05531545653893, 12.47771419321383

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