Properties

Label 2-1386-1.1-c3-0-35
Degree $2$
Conductor $1386$
Sign $1$
Analytic cond. $81.7766$
Root an. cond. $9.04304$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 14·5-s − 7·7-s + 8·8-s + 28·10-s − 11·11-s + 38·13-s − 14·14-s + 16·16-s − 54·17-s + 40·19-s + 56·20-s − 22·22-s − 8·23-s + 71·25-s + 76·26-s − 28·28-s + 170·29-s + 92·31-s + 32·32-s − 108·34-s − 98·35-s + 294·37-s + 80·38-s + 112·40-s + 258·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.25·5-s − 0.377·7-s + 0.353·8-s + 0.885·10-s − 0.301·11-s + 0.810·13-s − 0.267·14-s + 1/4·16-s − 0.770·17-s + 0.482·19-s + 0.626·20-s − 0.213·22-s − 0.0725·23-s + 0.567·25-s + 0.573·26-s − 0.188·28-s + 1.08·29-s + 0.533·31-s + 0.176·32-s − 0.544·34-s − 0.473·35-s + 1.30·37-s + 0.341·38-s + 0.442·40-s + 0.982·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(81.7766\)
Root analytic conductor: \(9.04304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.591005222\)
\(L(\frac12)\) \(\approx\) \(4.591005222\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
7 \( 1 + p T \)
11 \( 1 + p T \)
good5 \( 1 - 14 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 + 54 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 + 8 T + p^{3} T^{2} \)
29 \( 1 - 170 T + p^{3} T^{2} \)
31 \( 1 - 92 T + p^{3} T^{2} \)
37 \( 1 - 294 T + p^{3} T^{2} \)
41 \( 1 - 258 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 - 76 T + p^{3} T^{2} \)
53 \( 1 - 322 T + p^{3} T^{2} \)
59 \( 1 + 260 T + p^{3} T^{2} \)
61 \( 1 - 22 T + p^{3} T^{2} \)
67 \( 1 + 436 T + p^{3} T^{2} \)
71 \( 1 - 368 T + p^{3} T^{2} \)
73 \( 1 + 2 T + p^{3} T^{2} \)
79 \( 1 + 200 T + p^{3} T^{2} \)
83 \( 1 - 952 T + p^{3} T^{2} \)
89 \( 1 - 70 T + p^{3} T^{2} \)
97 \( 1 + 1086 T + p^{3} T^{2} \)
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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

−9.324245631410623736347659563994, −8.459931870195348166878874576035, −7.39975318403041806765718851409, −6.33096089891552510457493782868, −6.04983059237703409190300319664, −5.07585906373371268378362870029, −4.15285901027199055127791576529, −2.97195014472816331651961715846, −2.19577459424050434072290453825, −0.995473766092741112284988907525, 0.995473766092741112284988907525, 2.19577459424050434072290453825, 2.97195014472816331651961715846, 4.15285901027199055127791576529, 5.07585906373371268378362870029, 6.04983059237703409190300319664, 6.33096089891552510457493782868, 7.39975318403041806765718851409, 8.459931870195348166878874576035, 9.324245631410623736347659563994

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