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Properties

Label 2-2352-7.4-c1-0-1
Degree $2$
Conductor $2352$
Sign $-0.947 + 0.318i$
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 + (1.70 + 2.95i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s − 2.58·13-s − 3.41·15-s + (−1.12 + 1.94i)17-s + (1.41 + 2.44i)19-s + (−3.82 − 6.63i)23-s + (−3.32 + 5.76i)25-s + 0.999·27-s − 6.82·29-s + (0.585 − 1.01i)31-s + (−0.999 − 1.73i)33-s + (2 + 3.46i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.763 + 1.32i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s − 0.717·13-s − 0.881·15-s + (−0.271 + 0.471i)17-s + (0.324 + 0.561i)19-s + (−0.798 − 1.38i)23-s + (−0.665 + 1.15i)25-s + 0.192·27-s − 1.26·29-s + (0.105 − 0.182i)31-s + (−0.174 − 0.301i)33-s + (0.328 + 0.569i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7826825498\)
\(L(\frac12)\) \(\approx\) \(0.7826825498\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 2.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 + (1.12 - 1.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.82 + 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + (-0.585 + 1.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.585 + 1.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.12 - 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.82 - 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 + (-6.94 + 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.82 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 + (7.12 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−9.813279528455887288189137822859, −8.722800178107585595780319189034, −7.75789300821929654610196035580, −6.95819867207953408081140516977, −6.29576108767144463025188781868, −5.59923126544920025429837741171, −4.66046665014611978795504974966, −3.69780783483002676894171138462, −2.69187251322358883402885896332, −1.92492230425792169912528662613, 0.25676550043747858011932985074, 1.47381991701470758275874543860, 2.34750664456240727557624162190, 3.66798169541418190122368433306, 4.95220607381280058497057720362, 5.30715942788641762204539344097, 6.04955579194452940198268365749, 7.06391596930339478492291804476, 7.81584983383766933112253633457, 8.579086840569816273382183709269

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