Properties

Label 2-7168-1.1-c1-0-109
Degree $2$
Conductor $7168$
Sign $1$
Analytic cond. $57.2367$
Root an. cond. $7.56549$
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  + 2.88·3-s + 0.992·5-s + 7-s + 5.33·9-s − 3.42·11-s + 2.77·13-s + 2.86·15-s + 6.93·17-s − 1.96·19-s + 2.88·21-s − 2.05·23-s − 4.01·25-s + 6.73·27-s + 7.55·29-s − 5.23·31-s − 9.87·33-s + 0.992·35-s + 9.31·37-s + 8.00·39-s + 0.949·41-s − 8.41·43-s + 5.29·45-s + 4.64·47-s + 49-s + 20.0·51-s + 10.2·53-s − 3.39·55-s + ⋯
L(s)  = 1  + 1.66·3-s + 0.443·5-s + 0.377·7-s + 1.77·9-s − 1.03·11-s + 0.769·13-s + 0.739·15-s + 1.68·17-s − 0.450·19-s + 0.629·21-s − 0.428·23-s − 0.803·25-s + 1.29·27-s + 1.40·29-s − 0.940·31-s − 1.71·33-s + 0.167·35-s + 1.53·37-s + 1.28·39-s + 0.148·41-s − 1.28·43-s + 0.789·45-s + 0.677·47-s + 0.142·49-s + 2.80·51-s + 1.40·53-s − 0.457·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7168\)    =    \(2^{10} \cdot 7\)
Sign: $1$
Analytic conductor: \(57.2367\)
Root analytic conductor: \(7.56549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7168} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7168,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.743275560\)
\(L(\frac12)\) \(\approx\) \(4.743275560\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 - 2.88T + 3T^{2} \)
5 \( 1 - 0.992T + 5T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 - 6.93T + 17T^{2} \)
19 \( 1 + 1.96T + 19T^{2} \)
23 \( 1 + 2.05T + 23T^{2} \)
29 \( 1 - 7.55T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 - 9.31T + 37T^{2} \)
41 \( 1 - 0.949T + 41T^{2} \)
43 \( 1 + 8.41T + 43T^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 3.98T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 - 6.53T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 6.35T + 83T^{2} \)
89 \( 1 + 0.428T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−8.056576538797276145113400582737, −7.62867213854403840020682682556, −6.64843909354044092389492424963, −5.77367719986036349729151532887, −5.09440840997225862451070105803, −4.06106388817616657758920001446, −3.47864188610844268291059619739, −2.65159163783673107073952631272, −2.05122393751653222900864966016, −1.07983927757152369005050847453, 1.07983927757152369005050847453, 2.05122393751653222900864966016, 2.65159163783673107073952631272, 3.47864188610844268291059619739, 4.06106388817616657758920001446, 5.09440840997225862451070105803, 5.77367719986036349729151532887, 6.64843909354044092389492424963, 7.62867213854403840020682682556, 8.056576538797276145113400582737

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