Properties

Label 2-7168-1.1-c1-0-40
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  − 1.67·3-s − 2.65·5-s + 7-s − 0.202·9-s + 0.826·11-s + 5.57·13-s + 4.44·15-s − 1.74·17-s + 5.93·19-s − 1.67·21-s + 3.04·23-s + 2.05·25-s + 5.35·27-s + 6.26·29-s − 7.90·31-s − 1.38·33-s − 2.65·35-s − 8.30·37-s − 9.32·39-s + 1.38·41-s + 2.45·43-s + 0.538·45-s − 1.80·47-s + 49-s + 2.92·51-s + 13.7·53-s − 2.19·55-s + ⋯
L(s)  = 1  − 0.965·3-s − 1.18·5-s + 0.377·7-s − 0.0675·9-s + 0.249·11-s + 1.54·13-s + 1.14·15-s − 0.424·17-s + 1.36·19-s − 0.364·21-s + 0.634·23-s + 0.411·25-s + 1.03·27-s + 1.16·29-s − 1.42·31-s − 0.240·33-s − 0.448·35-s − 1.36·37-s − 1.49·39-s + 0.216·41-s + 0.375·43-s + 0.0802·45-s − 0.262·47-s + 0.142·49-s + 0.409·51-s + 1.89·53-s − 0.296·55-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

\(L(1)\) \(\approx\) \(1.087474147\)
\(L(\frac12)\) \(\approx\) \(1.087474147\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 + 2.65T + 5T^{2} \)
11 \( 1 - 0.826T + 11T^{2} \)
13 \( 1 - 5.57T + 13T^{2} \)
17 \( 1 + 1.74T + 17T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 - 3.04T + 23T^{2} \)
29 \( 1 - 6.26T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + 8.30T + 37T^{2} \)
41 \( 1 - 1.38T + 41T^{2} \)
43 \( 1 - 2.45T + 43T^{2} \)
47 \( 1 + 1.80T + 47T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 - 6.70T + 59T^{2} \)
61 \( 1 + 4.38T + 61T^{2} \)
67 \( 1 + 6.80T + 67T^{2} \)
71 \( 1 - 1.11T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 7.61T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 0.428T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−7.88774530006038618085730150497, −7.12838348778239053777313567444, −6.61878326801832987258614765951, −5.65667122841242020049716245639, −5.27072031779691892116666823750, −4.31051193487135021457145342801, −3.70119920741239347240779111250, −2.93381585435001037370060836696, −1.43847884252333167838144276985, −0.60695176692492363699790233119, 0.60695176692492363699790233119, 1.43847884252333167838144276985, 2.93381585435001037370060836696, 3.70119920741239347240779111250, 4.31051193487135021457145342801, 5.27072031779691892116666823750, 5.65667122841242020049716245639, 6.61878326801832987258614765951, 7.12838348778239053777313567444, 7.88774530006038618085730150497

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