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Properties

Label 2-1368-152.75-c1-0-13
Degree $2$
Conductor $1368$
Sign $-0.208 + 0.978i$
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  + (0.134 + 1.40i)2-s + (−1.96 + 0.378i)4-s + 2.61i·5-s + 4.81i·7-s + (−0.797 − 2.71i)8-s + (−3.68 + 0.351i)10-s − 3.49·11-s − 1.50·13-s + (−6.77 + 0.647i)14-s + (3.71 − 1.48i)16-s − 5.31·17-s + (3.83 + 2.07i)19-s + (−0.990 − 5.13i)20-s + (−0.470 − 4.91i)22-s − 2.26i·23-s + ⋯
L(s)  = 1  + (0.0951 + 0.995i)2-s + (−0.981 + 0.189i)4-s + 1.16i·5-s + 1.81i·7-s + (−0.281 − 0.959i)8-s + (−1.16 + 0.111i)10-s − 1.05·11-s − 0.416·13-s + (−1.81 + 0.173i)14-s + (0.928 − 0.371i)16-s − 1.28·17-s + (0.879 + 0.475i)19-s + (−0.221 − 1.14i)20-s + (−0.100 − 1.04i)22-s − 0.472i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.208 + 0.978i$
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: \((2,\ 1368,\ (\ :1/2),\ -0.208 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8127623970\)
\(L(\frac12)\) \(\approx\) \(0.8127623970\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.134 - 1.40i)T \)
3 \( 1 \)
19 \( 1 + (-3.83 - 2.07i)T \)
good5 \( 1 - 2.61iT - 5T^{2} \)
7 \( 1 - 4.81iT - 7T^{2} \)
11 \( 1 + 3.49T + 11T^{2} \)
13 \( 1 + 1.50T + 13T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 - 5.12T + 29T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 3.90iT - 41T^{2} \)
43 \( 1 - 0.342T + 43T^{2} \)
47 \( 1 + 0.679iT - 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 9.15iT - 59T^{2} \)
61 \( 1 - 2.65iT - 61T^{2} \)
67 \( 1 - 7.37iT - 67T^{2} \)
71 \( 1 - 9.74T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 - 4.80T + 83T^{2} \)
89 \( 1 + 14.0iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−9.956388919144610978204690091872, −9.203937209633056807394116162684, −8.432005906799857704817298792673, −7.70380117811649990515913421896, −6.81460182917010166742636536359, −6.08214703914577988873867045818, −5.41639204330816261941312727749, −4.52030898228526967886491985772, −3.01799387310680958387128931518, −2.44890111108631497530551726246, 0.34324286071651979580672888082, 1.28521594725658823333500563695, 2.69779637879886412860553618339, 3.87317076504343793614766785952, 4.74009287946303701632642752949, 5.05126238053880634225628268603, 6.54997810513459295574232462240, 7.72975175635106854751639797807, 8.180766105003679950240754440461, 9.408190681806259829302729604692

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