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Properties

Label 2-2352-12.11-c1-0-61
Degree $2$
Conductor $2352$
Sign $0.375 + 0.926i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.06 − 1.36i)3-s + 2.12i·5-s + (−0.732 − 2.90i)9-s + 5.81·11-s − 4.19·13-s + (2.90 + 2.26i)15-s − 5.81i·17-s − 2.73i·19-s − 4.25·23-s + 0.464·25-s + (−4.75 − 2.09i)27-s − 5.81i·29-s + 2.53i·31-s + (6.19 − 7.94i)33-s + 11.4·37-s + ⋯
L(s)  = 1  + (0.614 − 0.788i)3-s + 0.952i·5-s + (−0.244 − 0.969i)9-s + 1.75·11-s − 1.16·13-s + (0.751 + 0.585i)15-s − 1.41i·17-s − 0.626i·19-s − 0.888·23-s + 0.0928·25-s + (−0.914 − 0.403i)27-s − 1.08i·29-s + 0.455i·31-s + (1.07 − 1.38i)33-s + 1.88·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.375 + 0.926i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.375 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.205794787\)
\(L(\frac12)\) \(\approx\) \(2.205794787\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 1.36i)T \)
7 \( 1 \)
good5 \( 1 - 2.12iT - 5T^{2} \)
11 \( 1 - 5.81T + 11T^{2} \)
13 \( 1 + 4.19T + 13T^{2} \)
17 \( 1 + 5.81iT - 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + 4.25T + 23T^{2} \)
29 \( 1 + 5.81iT - 29T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 - 1.55iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 1.55iT - 53T^{2} \)
59 \( 1 - 9.50T + 59T^{2} \)
61 \( 1 - 1.26T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 - 9.50T + 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 - 4.92T + 97T^{2} \)
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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

−8.953058151019179786550443817807, −7.84762348019010578565168933634, −7.25841595078552854570383783946, −6.66491910158586219392696644156, −6.09952581088841582459131783015, −4.73453458479080426365760763200, −3.77833798921544425441279274454, −2.81598792922206112407964758780, −2.17549648294206041908367810959, −0.74599556850438012186290005260, 1.30371752356726278886608271162, 2.33953146811259799491245667448, 3.75605453047706339001234266262, 4.13033961439877710010153076726, 5.00178949716740210267453689080, 5.88271332604508245268825384290, 6.80866463100235009546682580267, 7.957186678559124745053198026291, 8.392645439462608595361364929561, 9.284149025485201067972920011057

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