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Properties

Label 2-2352-7.4-c1-0-8
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 + (0.414 − 0.717i)11-s − 4.24·13-s + 3.41·15-s + (−3.70 + 6.42i)17-s + (3.41 + 5.91i)19-s + (−2.41 − 4.18i)23-s + (−3.32 + 5.76i)25-s + 0.999·27-s + 2.82·29-s + (−1.41 + 2.44i)31-s + (0.414 + 0.717i)33-s + (0.828 + 1.43i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.763 − 1.32i)5-s + (−0.166 − 0.288i)9-s + (0.124 − 0.216i)11-s − 1.17·13-s + 0.881·15-s + (−0.899 + 1.55i)17-s + (0.783 + 1.35i)19-s + (−0.503 − 0.871i)23-s + (−0.665 + 1.15i)25-s + 0.192·27-s + 0.525·29-s + (−0.254 + 0.439i)31-s + (0.0721 + 0.124i)33-s + (0.136 + 0.235i)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

\(L(1)\) \(\approx\) \(1.029801835\)
\(L(\frac12)\) \(\approx\) \(1.029801835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (3.70 - 6.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.41 - 5.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.41 + 4.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (1.41 - 2.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.828 - 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (2.24 + 3.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.24 + 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.53 + 9.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.65 + 9.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (3.87 - 6.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.82 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (2.87 + 4.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.242T + 97T^{2} \)
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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

−9.023613087930906970033021186602, −8.187392140897132896705728632999, −7.83787874770112138449112185938, −6.59421162727527354545324478702, −5.74539201605017353001193258709, −4.92410591286054567394627194626, −4.25416431308752514173811881619, −3.62958049169295949889370623614, −2.11026284953365007849208036559, −0.74614105882130844943370644350, 0.55722235755361820308712145489, 2.49014090715462192335177686223, 2.79875017411983148017027612105, 4.13534364043385130907435545471, 4.93744783841756653642825520495, 5.96652619489415415363002759330, 6.94919332870422530627538578896, 7.34918530477277049329438137436, 7.67903064842454648546608497224, 9.108564555790496651175080940909

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