Properties

Label 2-2352-147.134-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.967 + 0.253i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 0.781i)3-s + (0.222 + 0.974i)7-s + (−0.222 + 0.974i)9-s + (−0.277 − 1.21i)13-s + 0.445·19-s + (0.623 − 0.781i)21-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + 1.80·31-s + (1.62 + 0.781i)37-s + (−0.777 + 0.974i)39-s + (0.277 − 0.347i)43-s + (−0.900 + 0.433i)49-s + (−0.277 − 0.347i)57-s + (1.62 + 0.781i)61-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)3-s + (0.222 + 0.974i)7-s + (−0.222 + 0.974i)9-s + (−0.277 − 1.21i)13-s + 0.445·19-s + (0.623 − 0.781i)21-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + 1.80·31-s + (1.62 + 0.781i)37-s + (−0.777 + 0.974i)39-s + (0.277 − 0.347i)43-s + (−0.900 + 0.433i)49-s + (−0.277 − 0.347i)57-s + (1.62 + 0.781i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9810123462\)
\(L(\frac12)\) \(\approx\) \(0.9810123462\)
\(L(1)\) 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 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
good5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 - 0.445T + T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 - 1.80T + T^{2} \)
37 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + 1.24T + T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 - 1.80T + T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−9.070912536179837772652979918384, −8.055701753817002127786677435889, −7.80520001814172966188010211259, −6.73191977137187420163090464040, −5.95307713932947613769477302519, −5.37796423260338461285520213676, −4.63435876105324579654945342183, −3.10143139114688824803716569439, −2.34216653912168372872321608434, −1.06123945941217682619177959914, 0.960962460042154568664335431842, 2.53617662027461825512937351681, 3.84229710000926466984221383408, 4.37111559939004713971391963156, 5.05255331506990352096867350174, 6.19055809328985292660186625024, 6.70649066623763126917224191507, 7.63952126258834031076049359560, 8.481860703345055961768587681375, 9.495623118733570686408290785513

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