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Properties

Label 2-1216-152.45-c1-0-18
Degree $2$
Conductor $1216$
Sign $0.999 + 0.0443i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (−1.80 + 1.04i)3-s + (−1.98 + 1.14i)5-s + 1.72·7-s + (0.675 − 1.17i)9-s − 2.75i·11-s + (−1.45 − 0.838i)13-s + (2.39 − 4.13i)15-s + (0.0573 + 0.0994i)17-s + (−3.55 − 2.52i)19-s + (−3.11 + 1.80i)21-s + (2.20 − 3.82i)23-s + (0.125 − 0.217i)25-s − 3.43i·27-s + (3.68 + 2.12i)29-s + 1.36·31-s + ⋯
L(s)  = 1  + (−1.04 + 0.602i)3-s + (−0.887 + 0.512i)5-s + 0.652·7-s + (0.225 − 0.390i)9-s − 0.830i·11-s + (−0.402 − 0.232i)13-s + (0.617 − 1.06i)15-s + (0.0139 + 0.0241i)17-s + (−0.815 − 0.578i)19-s + (−0.680 + 0.393i)21-s + (0.460 − 0.797i)23-s + (0.0251 − 0.0435i)25-s − 0.661i·27-s + (0.685 + 0.395i)29-s + 0.244·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.999 + 0.0443i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.999 + 0.0443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7928346993\)
\(L(\frac12)\) \(\approx\) \(0.7928346993\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (3.55 + 2.52i)T \)
good3 \( 1 + (1.80 - 1.04i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.98 - 1.14i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.72T + 7T^{2} \)
11 \( 1 + 2.75iT - 11T^{2} \)
13 \( 1 + (1.45 + 0.838i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.0573 - 0.0994i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.20 + 3.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.68 - 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.36T + 31T^{2} \)
37 \( 1 - 4.06iT - 37T^{2} \)
41 \( 1 + (0.293 + 0.507i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.28 + 0.743i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.00 + 5.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.48 - 4.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.0 + 6.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.2 - 5.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.72 - 2.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0279 + 0.0484i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.91 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.04 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.36iT - 83T^{2} \)
89 \( 1 + (-7.48 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.462 + 0.801i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.11784139705269206138474300942, −8.741728424266602794889814869962, −8.188312095321858783188776491799, −7.14990085485783091746289459667, −6.36497788085890061645325205264, −5.34139955984334708540839268317, −4.67240401634140705527693305926, −3.77686932251220839496661439604, −2.57480402015461775680846251357, −0.55733853052364200997895914422, 0.865714920456435802549219491760, 2.12912607980476620803560387792, 3.87649306391167744169734925422, 4.69873967604531736199322044282, 5.44950240202442236273592181874, 6.47742858401131477529398569294, 7.26949796838649986332131771743, 7.950380823623311917714049966051, 8.766179632918727170183029272473, 9.826826863999850893413740085807

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