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Properties

Label 2-3096-3096.1339-c0-0-1
Degree $2$
Conductor $3096$
Sign $0.130 + 0.991i$
Analytic cond. $1.54510$
Root an. cond. $1.24302$
Motivic weight $0$
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)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1 + 1.73i)11-s + (−0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (−0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s + (−0.999 − 1.73i)22-s + (−0.499 − 0.866i)24-s + 25-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1 + 1.73i)11-s + (−0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (−0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s + (−0.999 − 1.73i)22-s + (−0.499 − 0.866i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3096\)    =    \(2^{3} \cdot 3^{2} \cdot 43\)
Sign: $0.130 + 0.991i$
Analytic conductor: \(1.54510\)
Root analytic conductor: \(1.24302\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3096} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3096,\ (\ :0),\ 0.130 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4300762161\)
\(L(\frac12)\) \(\approx\) \(0.4300762161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.724876523393263512389567288399, −7.55336053557981365060673487987, −7.13298184048269113773440109123, −6.92527873473736275204953014493, −5.72658863371500179928435042836, −4.92261599394295215664092149347, −4.67068926485213146114969471313, −2.68637396590294861494121243729, −1.86630861845492519643255734076, −0.36182573572020555527100079761, 1.22474306445255035445638004533, 2.73665781343346058678421812821, 3.43521832338125289780514316345, 4.17570621589158379071686752234, 5.11982617414930839803218963800, 5.88424434486844322775281173351, 6.69381028053716571740745379781, 8.146156334417341975176479884374, 8.343570891270350070601214156103, 9.085979293388274316354240245533

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