Properties

Label 2-116242-1.1-c1-0-11
Degree $2$
Conductor $116242$
Sign $1$
Analytic cond. $928.197$
Root an. cond. $30.4663$
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 − 2·3-s + 4-s − 2·6-s + 7-s + 8-s + 9-s + 4·11-s − 2·12-s + 14-s + 16-s + 6·17-s + 18-s − 2·21-s + 4·22-s − 23-s − 2·24-s − 5·25-s + 4·27-s + 28-s − 10·29-s − 4·31-s + 32-s − 8·33-s + 6·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.436·21-s + 0.852·22-s − 0.208·23-s − 0.408·24-s − 25-s + 0.769·27-s + 0.188·28-s − 1.85·29-s − 0.718·31-s + 0.176·32-s − 1.39·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116242\)    =    \(2 \cdot 7 \cdot 19^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(928.197\)
Root analytic conductor: \(30.4663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{116242} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 116242,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.105072461\)
\(L(\frac12)\) \(\approx\) \(3.105072461\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.60435016168045, −13.01221197144261, −12.49561934896649, −12.03687376635614, −11.77351342602581, −11.30789051006659, −10.88646463470824, −10.42268814566147, −9.701884425704945, −9.359235555277191, −8.700025342424080, −7.977503063754188, −7.378715905770834, −7.168131091446374, −6.267476577415585, −5.965326788469143, −5.516182226871142, −5.195610939082102, −4.339266694266623, −3.897399624276942, −3.519727824262843, −2.583706108074849, −1.866502651297959, −1.218224278670188, −0.5573189488075701, 0.5573189488075701, 1.218224278670188, 1.866502651297959, 2.583706108074849, 3.519727824262843, 3.897399624276942, 4.339266694266623, 5.195610939082102, 5.516182226871142, 5.965326788469143, 6.267476577415585, 7.168131091446374, 7.378715905770834, 7.977503063754188, 8.700025342424080, 9.359235555277191, 9.701884425704945, 10.42268814566147, 10.88646463470824, 11.30789051006659, 11.77351342602581, 12.03687376635614, 12.49561934896649, 13.01221197144261, 13.60435016168045

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