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Properties

Label 2-1368-152.75-c1-0-94
Degree $2$
Conductor $1368$
Sign $-0.891 - 0.452i$
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  + (1.10 − 0.881i)2-s + (0.446 − 1.94i)4-s − 0.946i·5-s + 1.89i·7-s + (−1.22 − 2.54i)8-s + (−0.834 − 1.04i)10-s − 4.64·11-s − 4.52·13-s + (1.67 + 2.09i)14-s + (−3.60 − 1.73i)16-s − 4.07·17-s + (−3.46 + 2.64i)19-s + (−1.84 − 0.422i)20-s + (−5.13 + 4.08i)22-s − 5.34i·23-s + ⋯
L(s)  = 1  + (0.782 − 0.623i)2-s + (0.223 − 0.974i)4-s − 0.423i·5-s + 0.717i·7-s + (−0.433 − 0.901i)8-s + (−0.263 − 0.331i)10-s − 1.39·11-s − 1.25·13-s + (0.447 + 0.561i)14-s + (−0.900 − 0.434i)16-s − 0.989·17-s + (−0.794 + 0.607i)19-s + (−0.412 − 0.0944i)20-s + (−1.09 + 0.871i)22-s − 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.891 - 0.452i$
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: \((2,\ 1368,\ (\ :1/2),\ -0.891 - 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7442822431\)
\(L(\frac12)\) \(\approx\) \(0.7442822431\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 + 0.881i)T \)
3 \( 1 \)
19 \( 1 + (3.46 - 2.64i)T \)
good5 \( 1 + 0.946iT - 5T^{2} \)
7 \( 1 - 1.89iT - 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 + 4.07T + 17T^{2} \)
23 \( 1 + 5.34iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + 7.20T + 43T^{2} \)
47 \( 1 + 8.10iT - 47T^{2} \)
53 \( 1 - 4.02T + 53T^{2} \)
59 \( 1 - 9.43iT - 59T^{2} \)
61 \( 1 + 15.2iT - 61T^{2} \)
67 \( 1 - 9.68iT - 67T^{2} \)
71 \( 1 - 3.78T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 6.42T + 79T^{2} \)
83 \( 1 + 0.0483T + 83T^{2} \)
89 \( 1 - 3.55iT - 89T^{2} \)
97 \( 1 + 2.28iT - 97T^{2} \)
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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

−9.203282793344995257829121730820, −8.497585332444084681494817741043, −7.41631065368853779672356480168, −6.43766731428928384893386416638, −5.45717078162991442999846531984, −4.93182020824995065628934097548, −4.04316143061865368853493647997, −2.58064953190922498385931128077, −2.20237967516570797652522724308, −0.20447901675793402334906850630, 2.34112805793101540955776101584, 3.07422936849192424816941749825, 4.37633201772921651094384883973, 4.89982903729602357740500255861, 5.93029249823767157244647008972, 6.86822790168808877804721543479, 7.47664185411803570130481286345, 8.047070543345145329925778248881, 9.183573726022717319831733129241, 10.10169460619242386972132905783

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