Properties

Label 2-6930-1.1-c1-0-34
Degree $2$
Conductor $6930$
Sign $1$
Analytic cond. $55.3363$
Root an. cond. $7.43883$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s − 1.46·13-s − 14-s + 16-s + 3.46·17-s + 6.73·19-s + 20-s + 22-s + 8.19·23-s + 25-s + 1.46·26-s + 28-s + 4.73·29-s + 2·31-s − 32-s − 3.46·34-s + 35-s + 0.732·37-s − 6.73·38-s − 40-s + 2.19·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.406·13-s − 0.267·14-s + 0.250·16-s + 0.840·17-s + 1.54·19-s + 0.223·20-s + 0.213·22-s + 1.70·23-s + 0.200·25-s + 0.287·26-s + 0.188·28-s + 0.878·29-s + 0.359·31-s − 0.176·32-s − 0.594·34-s + 0.169·35-s + 0.120·37-s − 1.09·38-s − 0.158·40-s + 0.342·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(55.3363\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.876466479\)
\(L(\frac12)\) \(\approx\) \(1.876466479\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 0.732T + 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−7.80748736770738367921748501123, −7.46718747737599704874413811137, −6.72832019707687907676775785760, −5.84126860775113961707004879430, −5.21484201824838011314627972235, −4.55389300723680193371454084661, −3.16990213817617684112609492888, −2.78966160944805548774693460026, −1.56155159404533382987889960735, −0.841297465651754172922726154384, 0.841297465651754172922726154384, 1.56155159404533382987889960735, 2.78966160944805548774693460026, 3.16990213817617684112609492888, 4.55389300723680193371454084661, 5.21484201824838011314627972235, 5.84126860775113961707004879430, 6.72832019707687907676775785760, 7.46718747737599704874413811137, 7.80748736770738367921748501123

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