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Properties

Label 2-1368-152.75-c1-0-30
Degree $2$
Conductor $1368$
Sign $0.600 - 0.799i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
Motivic weight $1$
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 − 1.05i)2-s + (−0.233 + 1.98i)4-s + 1.70i·5-s + 4.15i·7-s + (2.31 − 1.61i)8-s + (1.80 − 1.60i)10-s + 2.55·11-s + 6.69·13-s + (4.39 − 3.90i)14-s + (−3.89 − 0.927i)16-s + 3.18·17-s + (−4.14 + 1.35i)19-s + (−3.39 − 0.398i)20-s + (−2.40 − 2.70i)22-s − 2.96i·23-s + ⋯
L(s)  = 1  + (−0.664 − 0.747i)2-s + (−0.116 + 0.993i)4-s + 0.763i·5-s + 1.57i·7-s + (0.819 − 0.572i)8-s + (0.570 − 0.507i)10-s + 0.771·11-s + 1.85·13-s + (1.17 − 1.04i)14-s + (−0.972 − 0.231i)16-s + 0.773·17-s + (−0.950 + 0.311i)19-s + (−0.758 − 0.0891i)20-s + (−0.512 − 0.576i)22-s − 0.618i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.293399109\)
\(L(\frac12)\) \(\approx\) \(1.293399109\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 1.05i)T \)
3 \( 1 \)
19 \( 1 + (4.14 - 1.35i)T \)
good5 \( 1 - 1.70iT - 5T^{2} \)
7 \( 1 - 4.15iT - 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
13 \( 1 - 6.69T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
23 \( 1 + 2.96iT - 23T^{2} \)
29 \( 1 + 5.33T + 29T^{2} \)
31 \( 1 + 2.28T + 31T^{2} \)
37 \( 1 - 4.67T + 37T^{2} \)
41 \( 1 - 2.45iT - 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 - 6.90iT - 47T^{2} \)
53 \( 1 + 0.126T + 53T^{2} \)
59 \( 1 + 0.727iT - 59T^{2} \)
61 \( 1 - 5.65iT - 61T^{2} \)
67 \( 1 + 3.07iT - 67T^{2} \)
71 \( 1 + 6.20T + 71T^{2} \)
73 \( 1 + 4.97T + 73T^{2} \)
79 \( 1 - 7.98T + 79T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + 8.04iT - 97T^{2} \)
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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

−9.609168821469179218567739591183, −8.877057157483422136308424200314, −8.483054496148836631070438136018, −7.47386374109236648410950811772, −6.31957496321443930006880367346, −5.86351229316782746113450875597, −4.26266664796026174852608179725, −3.35825485922842893143958374423, −2.50329428276264661825992440167, −1.40111972153795213002779424916, 0.792054461967069521823178595947, 1.50071282887527073607743628393, 3.75797017993669118974908651893, 4.30697538338814845596020585629, 5.51561315713350976346886631501, 6.30847962128849385348725241775, 7.12736075045883276341082158286, 7.85517679132141838283580128966, 8.702894401740339932490128250251, 9.223408840963689408387880531014

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