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Properties

Label 2-2352-1.1-c1-0-23
Degree $2$
Conductor $2352$
Sign $-1$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  − 3-s − 4·5-s + 9-s + 4·11-s − 4·13-s + 4·15-s + 4·19-s + 11·25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 6·37-s + 4·39-s − 4·43-s − 4·45-s − 8·47-s − 10·53-s − 16·55-s − 4·57-s + 4·59-s + 4·61-s + 16·65-s − 4·67-s − 8·71-s + 16·73-s − 11·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 1.03·15-s + 0.917·19-s + 11/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.640·39-s − 0.609·43-s − 0.596·45-s − 1.16·47-s − 1.37·53-s − 2.15·55-s − 0.529·57-s + 0.520·59-s + 0.512·61-s + 1.98·65-s − 0.488·67-s − 0.949·71-s + 1.87·73-s − 1.27·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.396588228342243005290523722036, −7.83338741005425949808014764620, −6.97102456634949378724083729434, −6.57921686914408075349576250571, −5.20899608454085443552291340826, −4.56539161642285599884271829131, −3.80836287756192507368269953204, −2.95291055879249797170654019370, −1.22910846835841870126165023591, 0, 1.22910846835841870126165023591, 2.95291055879249797170654019370, 3.80836287756192507368269953204, 4.56539161642285599884271829131, 5.20899608454085443552291340826, 6.57921686914408075349576250571, 6.97102456634949378724083729434, 7.83338741005425949808014764620, 8.396588228342243005290523722036

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