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Properties

Label 2-2352-28.3-c1-0-2
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 + (−1.45 + 0.842i)5-s + (−0.499 − 0.866i)9-s + (−3.05 − 1.76i)11-s − 2.93i·13-s + 1.68i·15-s + (−1.74 − 1.00i)17-s + (0.848 + 1.47i)19-s + (−1.38 + 0.799i)23-s + (−1.08 + 1.87i)25-s − 0.999·27-s + 7.94·29-s + (−2.47 + 4.29i)31-s + (−3.05 + 1.76i)33-s + (5.23 + 9.06i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.652 + 0.376i)5-s + (−0.166 − 0.288i)9-s + (−0.922 − 0.532i)11-s − 0.812i·13-s + 0.434i·15-s + (−0.422 − 0.243i)17-s + (0.194 + 0.337i)19-s + (−0.288 + 0.166i)23-s + (−0.216 + 0.374i)25-s − 0.192·27-s + 1.47·29-s + (−0.444 + 0.770i)31-s + (−0.532 + 0.307i)33-s + (0.860 + 1.49i)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: $\chi_{2352} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5524486205\)
\(L(\frac12)\) \(\approx\) \(0.5524486205\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.842i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.05 + 1.76i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + (1.74 + 1.00i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.848 - 1.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.38 - 0.799i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.94T + 29T^{2} \)
31 \( 1 + (2.47 - 4.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.23 - 9.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.86iT - 41T^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 + (3.35 + 5.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.46 - 2.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.87 - 6.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.9 - 6.29i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.47 + 0.850i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.13iT - 71T^{2} \)
73 \( 1 + (-2.10 - 1.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.749 + 0.432i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (13.8 - 7.96i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.82iT - 97T^{2} \)
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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

−9.068253919785720326399918104982, −8.069956500564048378355678462554, −7.947652207288348527732173158534, −7.00825439112596027920723092678, −6.20064182487796725360274031980, −5.33621224057418258453865562813, −4.37513825923040193090699842509, −3.15920951206141334807608372422, −2.79897667329551232545182983887, −1.25875952026350684171086454243, 0.18507156849684416969184536468, 1.97592914546316407465573648225, 2.90921816757539336261930693245, 4.15648812189873892037937912247, 4.48140175425800515620898864135, 5.44545127684101771654651482513, 6.45823899917880241282994531572, 7.39940692327154653690384424918, 8.045385698761308718410145525940, 8.724149102309382230745133796376

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