Properties

Label 2-6160-1.1-c1-0-45
Degree $2$
Conductor $6160$
Sign $1$
Analytic cond. $49.1878$
Root an. cond. $7.01340$
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  + 0.547·3-s + 5-s + 7-s − 2.70·9-s − 11-s + 6.76·13-s + 0.547·15-s + 1.45·17-s + 5.30·19-s + 0.547·21-s + 2.51·23-s + 25-s − 3.11·27-s + 2·29-s − 6.24·31-s − 0.547·33-s + 35-s + 1.48·37-s + 3.70·39-s + 7.15·41-s + 1.60·43-s − 2.70·45-s − 3.66·47-s + 49-s + 0.795·51-s + 4.70·53-s − 55-s + ⋯
L(s)  = 1  + 0.315·3-s + 0.447·5-s + 0.377·7-s − 0.900·9-s − 0.301·11-s + 1.87·13-s + 0.141·15-s + 0.352·17-s + 1.21·19-s + 0.119·21-s + 0.524·23-s + 0.200·25-s − 0.600·27-s + 0.371·29-s − 1.12·31-s − 0.0952·33-s + 0.169·35-s + 0.244·37-s + 0.592·39-s + 1.11·41-s + 0.245·43-s − 0.402·45-s − 0.534·47-s + 0.142·49-s + 0.111·51-s + 0.646·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6160\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(49.1878\)
Root analytic conductor: \(7.01340\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.784210411\)
\(L(\frac12)\) \(\approx\) \(2.784210411\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 - 0.547T + 3T^{2} \)
13 \( 1 - 6.76T + 13T^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 - 5.30T + 19T^{2} \)
23 \( 1 - 2.51T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 - 1.48T + 37T^{2} \)
41 \( 1 - 7.15T + 41T^{2} \)
43 \( 1 - 1.60T + 43T^{2} \)
47 \( 1 + 3.66T + 47T^{2} \)
53 \( 1 - 4.70T + 53T^{2} \)
59 \( 1 + 7.34T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 0.391T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 6.54T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + 5.18T + 89T^{2} \)
97 \( 1 + 6.21T + 97T^{2} \)
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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

−8.028272469475406656636231029300, −7.55670656188751965422802257876, −6.55060511279694125685635499850, −5.74981687156121929725538737935, −5.47346421047166254754131141281, −4.40197609676264434402413910551, −3.41735495456790402077140998510, −2.93037725275282670420591751759, −1.79855385240207087440330348550, −0.904499328125522667844029867054, 0.904499328125522667844029867054, 1.79855385240207087440330348550, 2.93037725275282670420591751759, 3.41735495456790402077140998510, 4.40197609676264434402413910551, 5.47346421047166254754131141281, 5.74981687156121929725538737935, 6.55060511279694125685635499850, 7.55670656188751965422802257876, 8.028272469475406656636231029300

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