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Properties

Label 2-1368-152.75-c1-0-67
Degree $2$
Conductor $1368$
Sign $0.820 - 0.571i$
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  + (1.39 + 0.244i)2-s + (1.88 + 0.680i)4-s + 3.10i·5-s − 4.34i·7-s + (2.45 + 1.40i)8-s + (−0.757 + 4.32i)10-s + 2.65·11-s + 5.10·13-s + (1.06 − 6.05i)14-s + (3.07 + 2.55i)16-s − 3.48·17-s + (1.86 − 3.93i)19-s + (−2.11 + 5.83i)20-s + (3.70 + 0.648i)22-s + 5.31i·23-s + ⋯
L(s)  = 1  + (0.984 + 0.172i)2-s + (0.940 + 0.340i)4-s + 1.38i·5-s − 1.64i·7-s + (0.867 + 0.497i)8-s + (−0.239 + 1.36i)10-s + 0.800·11-s + 1.41·13-s + (0.283 − 1.61i)14-s + (0.768 + 0.639i)16-s − 0.846·17-s + (0.427 − 0.903i)19-s + (−0.471 + 1.30i)20-s + (0.788 + 0.138i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.820 - 0.571i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.820 - 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.524049831\)
\(L(\frac12)\) \(\approx\) \(3.524049831\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.244i)T \)
3 \( 1 \)
19 \( 1 + (-1.86 + 3.93i)T \)
good5 \( 1 - 3.10iT - 5T^{2} \)
7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 - 5.10T + 13T^{2} \)
17 \( 1 + 3.48T + 17T^{2} \)
23 \( 1 - 5.31iT - 23T^{2} \)
29 \( 1 + 7.98T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 4.57T + 37T^{2} \)
41 \( 1 + 4.51iT - 41T^{2} \)
43 \( 1 - 6.28T + 43T^{2} \)
47 \( 1 - 7.68iT - 47T^{2} \)
53 \( 1 - 9.81T + 53T^{2} \)
59 \( 1 + 4.94iT - 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 - 0.406iT - 67T^{2} \)
71 \( 1 - 2.40T + 71T^{2} \)
73 \( 1 - 0.373T + 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 - 1.95iT - 89T^{2} \)
97 \( 1 + 4.87iT - 97T^{2} \)
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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

−9.955672462334211988895448228840, −8.786526279052014335942863528728, −7.46377124738851177668463668871, −7.13394726473567810381999161402, −6.49830194562240282570294315442, −5.64819036167416764050359414604, −4.16775851014218977272886308631, −3.81280785325137169130913038812, −2.90577669698425165787572568022, −1.44086539005799149431626904827, 1.31649710187331195737284915829, 2.25723224073935776211911298997, 3.61304798445028935448276926249, 4.41971580103753309534212461000, 5.40424360221490030242181085996, 5.88682736732937042540041451308, 6.67553065110653149801317698198, 8.114676299676614065705712327042, 8.793400754448080454215348330637, 9.257101963325113252861457790400

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