Properties

Label 2-236992-1.1-c1-0-5
Degree $2$
Conductor $236992$
Sign $1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
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  − 2·3-s + 2·5-s − 7-s + 9-s + 2·11-s − 4·13-s − 4·15-s + 2·17-s + 2·21-s − 25-s + 4·27-s + 2·29-s − 4·33-s − 2·35-s − 4·37-s + 8·39-s + 6·41-s + 2·43-s + 2·45-s + 4·47-s + 49-s − 4·51-s + 4·55-s − 2·59-s − 10·61-s − 63-s − 8·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 1.03·15-s + 0.485·17-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.696·33-s − 0.338·35-s − 0.657·37-s + 1.28·39-s + 0.937·41-s + 0.304·43-s + 0.298·45-s + 0.583·47-s + 1/7·49-s − 0.560·51-s + 0.539·55-s − 0.260·59-s − 1.28·61-s − 0.125·63-s − 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7861863683\)
\(L(\frac12)\) \(\approx\) \(0.7861863683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−12.76738178982407, −12.32001133585901, −12.03638506391928, −11.59968702162450, −11.05389304283937, −10.50864685573283, −10.17467897796773, −9.777708377470446, −9.174135793524744, −8.970384387683206, −8.148640836033411, −7.549725090940796, −7.108719717990460, −6.553903487480450, −6.153232034648766, −5.719288239668166, −5.341561913044803, −4.794675128286996, −4.270167225696846, −3.649277881018429, −2.775135130076398, −2.538386373748719, −1.602821802005707, −1.156628765831353, −0.2769154495420526, 0.2769154495420526, 1.156628765831353, 1.602821802005707, 2.538386373748719, 2.775135130076398, 3.649277881018429, 4.270167225696846, 4.794675128286996, 5.341561913044803, 5.719288239668166, 6.153232034648766, 6.553903487480450, 7.108719717990460, 7.549725090940796, 8.148640836033411, 8.970384387683206, 9.174135793524744, 9.777708377470446, 10.17467897796773, 10.50864685573283, 11.05389304283937, 11.59968702162450, 12.03638506391928, 12.32001133585901, 12.76738178982407

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