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Properties

Label 2-1368-1368.59-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.150 + 0.988i$
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  + (0.939 − 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s − 6-s + (0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s − 0.684i·11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)16-s + (0.939 + 0.342i)18-s + (−0.173 − 0.984i)19-s + (−0.233 − 0.642i)22-s + (−0.766 + 0.642i)24-s + (0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s − 6-s + (0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s − 0.684i·11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)16-s + (0.939 + 0.342i)18-s + (−0.173 − 0.984i)19-s + (−0.233 − 0.642i)22-s + (−0.766 + 0.642i)24-s + (0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.150 + 0.988i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.150 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.442384422\)
\(L(\frac12)\) \(\approx\) \(1.442384422\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
good5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + 0.684iT - T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
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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

−9.970031995357624874598958487674, −8.859912192894589656635933497594, −7.68772870993097268926440412529, −6.78782047437833212056569685717, −6.27098791822153467169276984167, −5.28047599475956335546549378422, −4.74557935608395493673151184842, −3.59545070219792478914290095412, −2.45604023922574924976838689234, −1.10350110481387059667592179699, 1.80129896114823085287066034617, 3.27826402835457662213222772579, 4.24756089958659077891570123452, 4.93296328195058961177954436037, 5.75280395955028449468856491017, 6.52783148661944558154857315722, 7.21051533946065495669638714581, 8.126119639914055054434251852605, 9.234972554787684811174412518335, 10.27880514618393272207230992769

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