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Properties

Label 2-1368-1368.1051-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.999 + 0.0151i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
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  + (0.173 + 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + 0.999·6-s + (−0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + 0.347·11-s + (0.173 + 0.984i)12-s + (0.766 − 0.642i)16-s + (1.53 − 1.28i)17-s + (0.173 − 0.984i)18-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)24-s + (0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + 0.999·6-s + (−0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + 0.347·11-s + (0.173 + 0.984i)12-s + (0.766 − 0.642i)16-s + (1.53 − 1.28i)17-s + (0.173 − 0.984i)18-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.999 + 0.0151i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.999 + 0.0151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131444806\)
\(L(\frac12)\) \(\approx\) \(1.131444806\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - 0.347T + T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.443629696795165273346707476090, −8.823573164770840090894746268318, −7.73725329280311348270177980507, −7.54365459758674680210962501132, −6.56447087231669983903544328837, −5.79611558385614359906262143754, −5.06567192777839128786242979895, −3.74526455105609671537830422025, −2.80071186756687473897408780898, −1.04753531556126707889247878159, 1.49456594073301148914440252572, 3.00728146216149728087787171911, 3.55567064592861768381024335324, 4.53601154462552804002952560531, 5.35104076511806941279427873281, 6.11557363986615151413172406764, 7.68033087247821868691752473478, 8.462177192198737962040221772794, 9.330866915035478158132737236581, 9.798100355156403102898937102232

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