Properties

Label 2-7168-1.1-c1-0-53
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  − 1.78·3-s − 4.18·5-s + 7-s + 0.189·9-s − 4.50·11-s − 4.84·13-s + 7.46·15-s + 5.13·17-s + 2.12·19-s − 1.78·21-s − 7.11·23-s + 12.4·25-s + 5.01·27-s − 5.43·29-s − 0.831·31-s + 8.04·33-s − 4.18·35-s + 7.98·37-s + 8.65·39-s − 2.22·41-s + 2.28·43-s − 0.790·45-s + 7.83·47-s + 49-s − 9.17·51-s + 7.90·53-s + 18.8·55-s + ⋯
L(s)  = 1  − 1.03·3-s − 1.87·5-s + 0.377·7-s + 0.0630·9-s − 1.35·11-s − 1.34·13-s + 1.92·15-s + 1.24·17-s + 0.488·19-s − 0.389·21-s − 1.48·23-s + 2.49·25-s + 0.966·27-s − 1.01·29-s − 0.149·31-s + 1.40·33-s − 0.706·35-s + 1.31·37-s + 1.38·39-s − 0.347·41-s + 0.348·43-s − 0.117·45-s + 1.14·47-s + 0.142·49-s − 1.28·51-s + 1.08·53-s + 2.53·55-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 \( 1 + 1.78T + 3T^{2} \)
5 \( 1 + 4.18T + 5T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 - 5.13T + 17T^{2} \)
19 \( 1 - 2.12T + 19T^{2} \)
23 \( 1 + 7.11T + 23T^{2} \)
29 \( 1 + 5.43T + 29T^{2} \)
31 \( 1 + 0.831T + 31T^{2} \)
37 \( 1 - 7.98T + 37T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 - 7.83T + 47T^{2} \)
53 \( 1 - 7.90T + 53T^{2} \)
59 \( 1 + 2.62T + 59T^{2} \)
61 \( 1 + 2.33T + 61T^{2} \)
67 \( 1 - 8.16T + 67T^{2} \)
71 \( 1 + 6.04T + 71T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 + 1.90T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 + 1.98T + 97T^{2} \)
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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

−7.61035274277095662743786528907, −7.21890624662563210641073361796, −5.98269034997491336391626799169, −5.38361781741740291221340843462, −4.80941134544095011238036838274, −4.11567727826782500940775086488, −3.23857899305892725059096592516, −2.39059124449902179758902319077, −0.75134698286118950310473968709, 0, 0.75134698286118950310473968709, 2.39059124449902179758902319077, 3.23857899305892725059096592516, 4.11567727826782500940775086488, 4.80941134544095011238036838274, 5.38361781741740291221340843462, 5.98269034997491336391626799169, 7.21890624662563210641073361796, 7.61035274277095662743786528907

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