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Properties

Label 2-2352-28.3-c1-0-6
Degree $2$
Conductor $2352$
Sign $-0.769 - 0.638i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.769 - 0.638i$
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: \((2,\ 2352,\ (\ :1/2),\ -0.769 - 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9334996362\)
\(L(\frac12)\) \(\approx\) \(0.9334996362\)
\(L(\frac{3}{2})\) 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

−9.201046761814364545269028294393, −8.750654662799850931478956912472, −7.65177744525766192156506355535, −6.97999493859283793944605160555, −6.15235216228819519467442274801, −5.28163432882705608563317232393, −4.48266001796376388663715744078, −3.68391186840654328555165093874, −2.72026143024608748588350629262, −1.35244767761032014593947089619, 0.34695404653939249825214405539, 1.57816748932413924067744591539, 2.82389149292745278358306307615, 3.69448858859817415844306064825, 4.88906922357442083147486084242, 5.51374166366176000503449756373, 6.36281234763551724337410149044, 7.22831455639303399268168508720, 7.87338051052981623903720333152, 8.502589839105032534774682777776

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