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Properties

Label 2-2352-7.4-c1-0-35
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 + (2.41 − 4.18i)11-s + 1.41·13-s − 3.41·15-s + (3.12 − 5.40i)17-s + (0.585 + 1.01i)19-s + (−0.414 − 0.717i)23-s + (−3.32 + 5.76i)25-s − 0.999·27-s − 8.48·29-s + (5.41 − 9.37i)31-s + (−2.41 − 4.18i)33-s + (4.82 + 8.36i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.763 − 1.32i)5-s + (−0.166 − 0.288i)9-s + (0.727 − 1.26i)11-s + 0.392·13-s − 0.881·15-s + (0.757 − 1.31i)17-s + (0.134 + 0.232i)19-s + (−0.0863 − 0.149i)23-s + (−0.665 + 1.15i)25-s − 0.192·27-s − 1.57·29-s + (0.972 − 1.68i)31-s + (−0.420 − 0.727i)33-s + (0.793 + 1.37i)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

\(L(1)\) \(\approx\) \(1.566478844\)
\(L(\frac12)\) \(\approx\) \(1.566478844\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + (-3.12 + 5.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.585 - 1.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.414 + 0.717i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-5.41 + 9.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 8.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-0.585 - 1.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.65 - 8.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.41 + 9.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.94 - 5.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.82T + 71T^{2} \)
73 \( 1 + (1.53 - 2.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.82 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 + (7.36 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−8.583264144886267586904320139613, −7.937669179060864337803194908126, −7.38268365106598235293686058676, −6.17864009440852319800875803842, −5.58711735384328896084579164136, −4.52059642562896008147224934009, −3.77573623908459863994608266061, −2.84504034134538881827147299755, −1.28865276233010613789576678004, −0.57090065081152910983353566370, 1.68351690274507679317685930558, 2.84306523921557145390910104862, 3.78193264035025920091634380083, 4.12719774092742370212296332237, 5.39489012244325391868188000894, 6.38465079230180221056826616440, 7.11073599072264754451425632851, 7.69507128402199008326512859533, 8.515107882903656108640436832739, 9.444470468232752976426185177037

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