Properties

Label 2-10304-1.1-c1-0-14
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 7-s + 9-s − 2·11-s − 4·13-s − 4·15-s − 6·17-s + 4·19-s + 2·21-s + 23-s − 25-s + 4·27-s − 6·29-s + 4·31-s + 4·33-s − 2·35-s + 8·37-s + 8·39-s + 6·41-s + 6·43-s + 2·45-s + 8·47-s + 49-s + 12·51-s + 4·53-s − 4·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 1.03·15-s − 1.45·17-s + 0.917·19-s + 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.338·35-s + 1.31·37-s + 1.28·39-s + 0.937·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 1.68·51-s + 0.549·53-s − 0.539·55-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: \(1\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.97119318737866, −16.42402134916296, −15.87204950816534, −15.28796422559443, −14.60231571332903, −13.88192432366202, −13.37779484974007, −12.79268165325685, −12.32506624596113, −11.50295585158313, −11.17092011439315, −10.48360632462261, −9.847429727401846, −9.444008497353435, −8.747064649309010, −7.746161739579152, −7.132289407714452, −6.517443722032271, −5.737059273233733, −5.511436981254922, −4.722724316940364, −4.013862810732974, −2.614277410579654, −2.372075543050787, −0.9766673864714591, 0, 0.9766673864714591, 2.372075543050787, 2.614277410579654, 4.013862810732974, 4.722724316940364, 5.511436981254922, 5.737059273233733, 6.517443722032271, 7.132289407714452, 7.746161739579152, 8.747064649309010, 9.444008497353435, 9.847429727401846, 10.48360632462261, 11.17092011439315, 11.50295585158313, 12.32506624596113, 12.79268165325685, 13.37779484974007, 13.88192432366202, 14.60231571332903, 15.28796422559443, 15.87204950816534, 16.42402134916296, 16.97119318737866

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