Properties

Degree $2$
Conductor $348726$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

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 − 4·11-s + 12-s + 2·13-s − 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·20-s + 21-s + 4·22-s − 23-s − 24-s − 25-s − 2·26-s + 27-s + 28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-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 + 0.852·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-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: $1$
Motivic weight: \(1\)
Character: $\chi_{348726} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.17034741803910, −12.40773546961804, −12.06261623954654, −11.55206549819575, −11.10187983035216, −10.66647832976814, −10.44774750574568, −9.670529212732078, −9.416301021200528, −8.693296133370262, −8.333327230637152, −8.110636550702043, −7.709094212883558, −7.141799217607515, −6.666776307535916, −6.215231603106071, −5.549896766968571, −4.878579873320890, −4.441839865569226, −4.030510052660080, −3.258179555369229, −2.861708144860395, −2.339225618162108, −1.689855292237273, −1.131963019515052, 0, 0, 1.131963019515052, 1.689855292237273, 2.339225618162108, 2.861708144860395, 3.258179555369229, 4.030510052660080, 4.441839865569226, 4.878579873320890, 5.549896766968571, 6.215231603106071, 6.666776307535916, 7.141799217607515, 7.709094212883558, 8.110636550702043, 8.333327230637152, 8.693296133370262, 9.416301021200528, 9.670529212732078, 10.44774750574568, 10.66647832976814, 11.10187983035216, 11.55206549819575, 12.06261623954654, 12.40773546961804, 13.17034741803910

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