Properties

Label 2-348726-1.1-c1-0-0
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 + 6·11-s − 12-s − 6·13-s + 14-s + 2·15-s + 16-s − 2·17-s − 18-s − 2·20-s + 21-s − 6·22-s − 23-s + 24-s − 25-s + 6·26-s − 27-s − 28-s − 10·29-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 + 1.80·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s − 1.27·22-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 − 1.85·29-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: \(0\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02206995353\)
\(L(\frac12)\) \(\approx\) \(0.02206995353\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.27973331578003, −12.07546947849299, −11.50735451436492, −11.38753957093456, −10.87070199383996, −10.02327993507832, −9.877021585074950, −9.476876240670518, −8.963184392823448, −8.427874744485629, −7.983816243493778, −7.334191974885015, −7.061565996707366, −6.711946320992748, −6.142100735505099, −5.694146262583650, −4.863767154035141, −4.593571545906587, −3.913641316960724, −3.517810516139524, −2.955763934071615, −2.026102544978200, −1.737948628955166, −0.9187158595426046, −0.05352173160283814, 0.05352173160283814, 0.9187158595426046, 1.737948628955166, 2.026102544978200, 2.955763934071615, 3.517810516139524, 3.913641316960724, 4.593571545906587, 4.863767154035141, 5.694146262583650, 6.142100735505099, 6.711946320992748, 7.061565996707366, 7.334191974885015, 7.983816243493778, 8.427874744485629, 8.963184392823448, 9.476876240670518, 9.877021585074950, 10.02327993507832, 10.87070199383996, 11.38753957093456, 11.50735451436492, 12.07546947849299, 12.27973331578003

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