Properties

Label 2-7168-1.1-c1-0-76
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  − 2.29·3-s − 1.65·5-s + 7-s + 2.28·9-s − 6.22·11-s + 0.634·13-s + 3.80·15-s − 5.02·17-s − 2.11·19-s − 2.29·21-s + 8.89·23-s − 2.26·25-s + 1.63·27-s + 1.13·29-s + 8.27·31-s + 14.3·33-s − 1.65·35-s + 2.19·37-s − 1.45·39-s − 4.93·41-s + 5.27·43-s − 3.78·45-s + 6.68·47-s + 49-s + 11.5·51-s − 6.41·53-s + 10.2·55-s + ⋯
L(s)  = 1  − 1.32·3-s − 0.739·5-s + 0.377·7-s + 0.762·9-s − 1.87·11-s + 0.175·13-s + 0.981·15-s − 1.21·17-s − 0.486·19-s − 0.501·21-s + 1.85·23-s − 0.453·25-s + 0.315·27-s + 0.210·29-s + 1.48·31-s + 2.49·33-s − 0.279·35-s + 0.360·37-s − 0.233·39-s − 0.769·41-s + 0.805·43-s − 0.563·45-s + 0.975·47-s + 0.142·49-s + 1.61·51-s − 0.880·53-s + 1.38·55-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 \( 1 + 2.29T + 3T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
11 \( 1 + 6.22T + 11T^{2} \)
13 \( 1 - 0.634T + 13T^{2} \)
17 \( 1 + 5.02T + 17T^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
23 \( 1 - 8.89T + 23T^{2} \)
29 \( 1 - 1.13T + 29T^{2} \)
31 \( 1 - 8.27T + 31T^{2} \)
37 \( 1 - 2.19T + 37T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 - 5.27T + 43T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 4.07T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 - 3.70T + 83T^{2} \)
89 \( 1 - 1.60T + 89T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−7.46224728521480983837498200740, −6.87900226707225045730204761827, −6.11001819967909376250029183801, −5.40265705961496819187655476745, −4.74721349480177430879187278280, −4.37560962781428457275710937457, −3.07994620754489182834152130469, −2.31547647017265897355957252022, −0.870123745062703292528198465976, 0, 0.870123745062703292528198465976, 2.31547647017265897355957252022, 3.07994620754489182834152130469, 4.37560962781428457275710937457, 4.74721349480177430879187278280, 5.40265705961496819187655476745, 6.11001819967909376250029183801, 6.87900226707225045730204761827, 7.46224728521480983837498200740

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