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Properties

Label 2-2352-28.19-c1-0-11
Degree $2$
Conductor $2352$
Sign $0.327 - 0.944i$
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  + (−0.5 − 0.866i)3-s + (2.78 + 1.60i)5-s + (−0.499 + 0.866i)9-s + (−1.18 + 0.683i)11-s + 2.93i·13-s − 3.21i·15-s + (−5.98 + 3.45i)17-s + (3.67 − 6.36i)19-s + (3.14 + 1.81i)23-s + (2.66 + 4.62i)25-s + 0.999·27-s − 1.11·29-s + (4.35 + 7.53i)31-s + (1.18 + 0.683i)33-s + (−3.81 + 6.61i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (1.24 + 0.718i)5-s + (−0.166 + 0.288i)9-s + (−0.356 + 0.206i)11-s + 0.812i·13-s − 0.830i·15-s + (−1.45 + 0.838i)17-s + (0.843 − 1.46i)19-s + (0.655 + 0.378i)23-s + (0.533 + 0.924i)25-s + 0.192·27-s − 0.206·29-s + (0.781 + 1.35i)31-s + (0.206 + 0.118i)33-s + (−0.627 + 1.08i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.684522937\)
\(L(\frac12)\) \(\approx\) \(1.684522937\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.78 - 1.60i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.18 - 0.683i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.93iT - 13T^{2} \)
17 \( 1 + (5.98 - 3.45i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.67 + 6.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.14 - 1.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.11T + 29T^{2} \)
31 \( 1 + (-4.35 - 7.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.81 - 6.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.833iT - 41T^{2} \)
43 \( 1 - 4.82iT - 43T^{2} \)
47 \( 1 + (-1.47 + 2.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.28 - 3.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.04 + 12.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.57 - 5.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (-5.62 + 3.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.74 + 3.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.87T + 83T^{2} \)
89 \( 1 + (-10.9 - 6.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−9.115551495066217507043426492649, −8.519117496684053017381091494880, −7.23787479688264948332924651200, −6.76469035213985947484189993575, −6.24810574673794402767036025796, −5.25656098239929918974774710144, −4.55691336997470804062887837736, −3.06611282732966051831083467397, −2.30849322921343581710203366781, −1.39542168184524185670905075947, 0.58674529577124584020170950987, 1.93923127916025190827610127816, 2.92918950098814766413657101988, 4.13244190829624209463896513405, 5.05675731378429740874437636046, 5.60151313613875202807083408931, 6.20289677011915710080569525238, 7.28765174734726958434081506339, 8.221373968191706574096817366112, 9.068313080300168886624204198691

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