Properties

Label 2-7938-1.1-c1-0-144
Degree $2$
Conductor $7938$
Sign $-1$
Analytic cond. $63.3852$
Root an. cond. $7.96148$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 4.24·11-s + 2.24·13-s + 16-s + 2.24·19-s − 4.24·22-s − 1.24·23-s − 5·25-s + 2.24·26-s + 4.24·29-s − 9.24·31-s + 32-s − 4·37-s + 2.24·38-s − 11.4·41-s + 10.4·43-s − 4.24·44-s − 1.24·46-s − 4.75·47-s − 5·50-s + 2.24·52-s − 4.24·53-s + 4.24·58-s + 2.24·61-s − 9.24·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.353·8-s − 1.27·11-s + 0.621·13-s + 0.250·16-s + 0.514·19-s − 0.904·22-s − 0.259·23-s − 25-s + 0.439·26-s + 0.787·29-s − 1.66·31-s + 0.176·32-s − 0.657·37-s + 0.363·38-s − 1.79·41-s + 1.59·43-s − 0.639·44-s − 0.183·46-s − 0.693·47-s − 0.707·50-s + 0.310·52-s − 0.582·53-s + 0.557·58-s + 0.287·61-s − 1.17·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7938\)    =    \(2 \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(63.3852\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7938,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 9.24T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 4.75T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2.24T + 61T^{2} \)
67 \( 1 - 0.242T + 67T^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 4.48T + 97T^{2} \)
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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

−7.40971043247299713100666432346, −6.80202217732845795939464729903, −5.86477322032396701234545869710, −5.45608072161696427135646757883, −4.74922958778590638718419471749, −3.84922054848752280456182446681, −3.22570622191344067617966086721, −2.36391495039277332205872234668, −1.48756799661581509278497207182, 0, 1.48756799661581509278497207182, 2.36391495039277332205872234668, 3.22570622191344067617966086721, 3.84922054848752280456182446681, 4.74922958778590638718419471749, 5.45608072161696427135646757883, 5.86477322032396701234545869710, 6.80202217732845795939464729903, 7.40971043247299713100666432346

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