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Properties

Label 2-1368-1368.859-c0-0-0
Degree $2$
Conductor $1368$
Sign $-0.378 + 0.925i$
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 + (−0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−0.766 + 1.32i)11-s + (0.173 − 0.984i)12-s + (0.766 − 0.642i)16-s + (−1.87 − 0.684i)17-s + (0.766 − 0.642i)18-s + (−0.939 + 0.342i)19-s + (−1.43 − 0.524i)22-s + 0.999·24-s + (−0.939 + 0.342i)25-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−0.766 + 1.32i)11-s + (0.173 − 0.984i)12-s + (0.766 − 0.642i)16-s + (−1.87 − 0.684i)17-s + (0.766 − 0.642i)18-s + (−0.939 + 0.342i)19-s + (−1.43 − 0.524i)22-s + 0.999·24-s + (−0.939 + 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3443708033\)
\(L(\frac12)\) \(\approx\) \(0.3443708033\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 - 1.53T + T^{2} \)
89 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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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

−10.19884929831162974546686500935, −9.390259849240522267763564603313, −8.852446703593935934162251072908, −7.80133745228839099078454474672, −6.96591821606840412908047919435, −6.21693689657049994559068136518, −5.28874848635526302501875890500, −4.53483647896850311120643275300, −4.00667315732449319582489333184, −2.47993740918818229724802701289, 0.27112283062176131331796015235, 1.91448329070623219941270031189, 2.69635582228939759072176510442, 3.99275793384514625725584890439, 4.96588727201958137153427373380, 5.93907368015460238500615358177, 6.48180455460170379161220832893, 7.83899819123040920448136564942, 8.497733848829218322058313484326, 9.148925545010146224376032503324

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