Properties

Label 2-7168-1.1-c1-0-176
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.42·3-s − 3.59·5-s + 7-s + 8.70·9-s − 1.08·11-s − 1.79·13-s − 12.2·15-s − 5.65·17-s + 0.0630·19-s + 3.42·21-s + 1.46·23-s + 7.91·25-s + 19.5·27-s − 5.04·29-s − 4.75·31-s − 3.70·33-s − 3.59·35-s − 7.19·37-s − 6.12·39-s − 7.50·41-s − 4.56·43-s − 31.2·45-s − 1.52·47-s + 49-s − 19.3·51-s − 6.59·53-s + 3.88·55-s + ⋯
L(s)  = 1  + 1.97·3-s − 1.60·5-s + 0.377·7-s + 2.90·9-s − 0.326·11-s − 0.496·13-s − 3.17·15-s − 1.37·17-s + 0.0144·19-s + 0.746·21-s + 0.305·23-s + 1.58·25-s + 3.75·27-s − 0.936·29-s − 0.853·31-s − 0.644·33-s − 0.607·35-s − 1.18·37-s − 0.980·39-s − 1.17·41-s − 0.695·43-s − 4.66·45-s − 0.222·47-s + 0.142·49-s − 2.70·51-s − 0.905·53-s + 0.524·55-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: \(1\)
Selberg data: \((2,\ 7168,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good3 \( 1 - 3.42T + 3T^{2} \)
5 \( 1 + 3.59T + 5T^{2} \)
11 \( 1 + 1.08T + 11T^{2} \)
13 \( 1 + 1.79T + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 - 0.0630T + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + 5.04T + 29T^{2} \)
31 \( 1 + 4.75T + 31T^{2} \)
37 \( 1 + 7.19T + 37T^{2} \)
41 \( 1 + 7.50T + 41T^{2} \)
43 \( 1 + 4.56T + 43T^{2} \)
47 \( 1 + 1.52T + 47T^{2} \)
53 \( 1 + 6.59T + 53T^{2} \)
59 \( 1 + 7.62T + 59T^{2} \)
61 \( 1 - 9.62T + 61T^{2} \)
67 \( 1 + 6.97T + 67T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 - 8.59T + 73T^{2} \)
79 \( 1 - 7.84T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 9.32T + 89T^{2} \)
97 \( 1 - 0.485T + 97T^{2} \)
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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

−7.64142766165825586936636704002, −7.26136228926417976015751834112, −6.67103121284576146067065247192, −4.99090153559875556131048638393, −4.52703537737972636804473389287, −3.69252778866156833702089189640, −3.33686321213875451838276412709, −2.37760080028066918549294015102, −1.62481689224112741287277089158, 0, 1.62481689224112741287277089158, 2.37760080028066918549294015102, 3.33686321213875451838276412709, 3.69252778866156833702089189640, 4.52703537737972636804473389287, 4.99090153559875556131048638393, 6.67103121284576146067065247192, 7.26136228926417976015751834112, 7.64142766165825586936636704002

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