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Properties

Label 2-2352-12.11-c1-0-5
Degree $2$
Conductor $2352$
Sign $-i$
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  − 1.73i·3-s − 2.99·9-s − 7·13-s − 5.19i·19-s + 5·25-s + 5.19i·27-s + 8.66i·31-s − 37-s + 12.1i·39-s + 12.1i·43-s − 9·57-s − 14·61-s + 12.1i·67-s − 7·73-s − 8.66i·75-s + ⋯
L(s)  = 1  − 0.999i·3-s − 0.999·9-s − 1.94·13-s − 1.19i·19-s + 25-s + 0.999i·27-s + 1.55i·31-s − 0.164·37-s + 1.94i·39-s + 1.84i·43-s − 1.19·57-s − 1.79·61-s + 1.48i·67-s − 0.819·73-s − 0.999i·75-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-i$
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: \((2,\ 2352,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3978417265\)
\(L(\frac12)\) \(\approx\) \(0.3978417265\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.057388009487985982741926637639, −8.366364291848424234648865922414, −7.38269028665345657070837334659, −7.07186959440517697353335820752, −6.25360107049326898120551386870, −5.15218771369243878379535029525, −4.65571740471496100887281170939, −3.03048460560840127825364430557, −2.51263898296052585980890995263, −1.24460249660826580575757902951, 0.13523685308600440766328821257, 2.10559207402222168616955538853, 3.03115392900874834067879813453, 4.01026648894047029139493730596, 4.80144729616672893954543188408, 5.45255514103565750986780914797, 6.33533873425176335963614904931, 7.41954774703763249464139903326, 8.000943590282717289561775407830, 9.053035966969722108950545627598

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