Properties

Label 2-2352-28.3-c1-0-0
Degree $2$
Conductor $2352$
Sign $-0.995 + 0.0956i$
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 + (0.521 − 0.300i)5-s + (−0.499 − 0.866i)9-s + (0.141 + 0.0818i)11-s − 3.37i·13-s + 0.601i·15-s + (−1.84 − 1.06i)17-s + (−0.476 − 0.825i)19-s + (−6.05 + 3.49i)23-s + (−2.31 + 4.01i)25-s + 0.999·27-s − 1.28·29-s + (−1.67 + 2.90i)31-s + (−0.141 + 0.0818i)33-s + (−2.58 − 4.47i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.233 − 0.134i)5-s + (−0.166 − 0.288i)9-s + (0.0427 + 0.0246i)11-s − 0.937i·13-s + 0.155i·15-s + (−0.447 − 0.258i)17-s + (−0.109 − 0.189i)19-s + (−1.26 + 0.729i)23-s + (−0.463 + 0.803i)25-s + 0.192·27-s − 0.239·29-s + (−0.301 + 0.521i)31-s + (−0.0246 + 0.0142i)33-s + (−0.424 − 0.735i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.995 + 0.0956i$
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: \((2,\ 2352,\ (\ :1/2),\ -0.995 + 0.0956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1217695818\)
\(L(\frac12)\) \(\approx\) \(0.1217695818\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.521 + 0.300i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.141 - 0.0818i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + (1.84 + 1.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.476 + 0.825i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.05 - 3.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 + (1.67 - 2.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.58 + 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 - 6.09iT - 43T^{2} \)
47 \( 1 + (-1.84 - 3.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.94 - 12.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.02 - 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.157 - 0.0907i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.87 - 4.54i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.36iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.05 - 3.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + (-12.8 + 7.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.08iT - 97T^{2} \)
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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

−9.386824321738328167881501655236, −8.782467294518364642335395506651, −7.79203023132394746719151383185, −7.16131106985616561069378757206, −5.95555729823332966598477831716, −5.60208471546738390795178252604, −4.61347232760518851314170919978, −3.77042162462556543747280066459, −2.81450112749977755363737464265, −1.54772962202823722350763495384, 0.04052515659000684919115130168, 1.68790968745034430789287755213, 2.40322172954529848079046814969, 3.76970164453704748965358000753, 4.55620561287597243431581687270, 5.57973833493353722713743878268, 6.44723952712236182549545234036, 6.77183385445754750720004450089, 7.932021102483711326301077865647, 8.409966989538653394516627696326

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