Properties

Label 2-10304-1.1-c1-0-21
Degree $2$
Conductor $10304$
Sign $1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·5-s − 7-s + 6·9-s − 2·11-s − 5·13-s + 12·15-s + 4·19-s − 3·21-s + 23-s + 11·25-s + 9·27-s + 3·29-s + 5·31-s − 6·33-s − 4·35-s − 4·37-s − 15·39-s + 5·41-s + 4·43-s + 24·45-s − 11·47-s + 49-s − 8·55-s + 12·57-s + 12·59-s + 6·61-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.78·5-s − 0.377·7-s + 2·9-s − 0.603·11-s − 1.38·13-s + 3.09·15-s + 0.917·19-s − 0.654·21-s + 0.208·23-s + 11/5·25-s + 1.73·27-s + 0.557·29-s + 0.898·31-s − 1.04·33-s − 0.676·35-s − 0.657·37-s − 2.40·39-s + 0.780·41-s + 0.609·43-s + 3.57·45-s − 1.60·47-s + 1/7·49-s − 1.07·55-s + 1.58·57-s + 1.56·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.804408370\)
\(L(\frac12)\) \(\approx\) \(5.804408370\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−16.52296840526787, −15.93600517039981, −15.29071505978200, −14.68130768491984, −14.12815749303628, −13.92083928043469, −13.30353183341918, −12.77533686694304, −12.38662414996057, −11.26804200732754, −10.14561901439347, −10.07960432108480, −9.482018679074787, −9.143195682640308, −8.264940795792882, −7.806501113722858, −6.885030143357390, −6.578849385072158, −5.316099501555570, −5.116812538412009, −3.986007163726490, −3.007044013522224, −2.548706019070745, −2.115647349838402, −1.077427571647196, 1.077427571647196, 2.115647349838402, 2.548706019070745, 3.007044013522224, 3.986007163726490, 5.116812538412009, 5.316099501555570, 6.578849385072158, 6.885030143357390, 7.806501113722858, 8.264940795792882, 9.143195682640308, 9.482018679074787, 10.07960432108480, 10.14561901439347, 11.26804200732754, 12.38662414996057, 12.77533686694304, 13.30353183341918, 13.92083928043469, 14.12815749303628, 14.68130768491984, 15.29071505978200, 15.93600517039981, 16.52296840526787

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