Properties

Degree $2$
Conductor $2352$
Sign $0.947 - 0.318i$
Motivic weight $1$
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 + (−0.292 − 0.507i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 5.41·13-s − 0.585·15-s + (−3.12 + 5.40i)17-s + (1.41 + 2.44i)19-s + (1.82 + 3.16i)23-s + (2.32 − 4.03i)25-s − 0.999·27-s − 1.17·29-s + (−3.41 + 5.91i)31-s + (0.999 + 1.73i)33-s + (2 + 3.46i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.130 − 0.226i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 1.50·13-s − 0.151·15-s + (−0.757 + 1.31i)17-s + (0.324 + 0.561i)19-s + (0.381 + 0.660i)23-s + (0.465 − 0.806i)25-s − 0.192·27-s − 0.217·29-s + (−0.613 + 1.06i)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: $0.947 - 0.318i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(1.882532673\)
\(L(\frac12)\) \(\approx\) \(1.882532673\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 0.507i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 + (3.12 - 5.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.82 - 3.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (3.41 - 5.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
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 \( 1 + (3.41 - 5.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.87 + 3.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.82 + 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (-2.94 + 5.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.17 - 2.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + (-2.87 - 4.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.41T + 97T^{2} \)
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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

−8.827090754214502111043459927860, −8.292153954544744291944235580449, −7.61684821961654307893374423423, −6.62915027159913340096840531387, −6.08832715504166373699955198292, −5.09463882897299474946038088414, −4.03890435393439115046861694035, −3.33196213394815158410855379853, −2.05118661902886876074088602030, −1.16205599674845513647561804054, 0.70887067804965843296346965018, 2.32876643329338936231765442298, 3.21405128954750419942968365275, 3.99076366076338142621916970024, 4.95532838166331572820802489256, 5.74059126780307498440962963580, 6.66276971195713430996892567061, 7.43631226993604205221436051271, 8.302503271208564284487486545145, 9.069015916072700014569291689525

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