Properties

Label 2-1216-19.11-c1-0-35
Degree $2$
Conductor $1216$
Sign $-0.988 - 0.149i$
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  + (1.59 − 2.75i)3-s + (−1.28 + 2.21i)5-s + (−3.56 − 6.16i)9-s − 4.96·11-s + (−2.28 − 3.95i)13-s + (4.07 + 7.05i)15-s + (2.28 − 3.95i)17-s + (0.697 + 4.30i)19-s + (0.893 + 1.54i)23-s + (−0.780 − 1.35i)25-s − 13.1·27-s + (−3.84 − 6.65i)29-s − 6.36·31-s + (−7.90 + 13.6i)33-s − 7.12·37-s + ⋯
L(s)  = 1  + (0.918 − 1.59i)3-s + (−0.572 + 0.992i)5-s + (−1.18 − 2.05i)9-s − 1.49·11-s + (−0.632 − 1.09i)13-s + (1.05 + 1.82i)15-s + (0.553 − 0.958i)17-s + (0.160 + 0.987i)19-s + (0.186 + 0.322i)23-s + (−0.156 − 0.270i)25-s − 2.52·27-s + (−0.713 − 1.23i)29-s − 1.14·31-s + (−1.37 + 2.38i)33-s − 1.17·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8439793560\)
\(L(\frac12)\) \(\approx\) \(0.8439793560\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.697 - 4.30i)T \)
good3 \( 1 + (-1.59 + 2.75i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.28 - 2.21i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4.96T + 11T^{2} \)
13 \( 1 + (2.28 + 3.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.893 - 1.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.84 + 6.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.36T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.28 + 3.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.07 - 7.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.28 - 3.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.59 - 2.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.28 + 5.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.98 + 5.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.893 - 1.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.62 + 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.07 + 7.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.39T + 83T^{2} \)
89 \( 1 + (0.842 + 1.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.62 + 4.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.143386129579345738019726065183, −7.982994588246555438495820634611, −7.52243385027961905794233054693, −7.40888274395982272699923971814, −6.13845137454861106688741241767, −5.28474433843060335443091484820, −3.40523515343781593190503374611, −2.95511754868336794445868667760, −2.01251757137446784241719311740, −0.29212685209668569000213088047, 2.15639424872154195928723125206, 3.29392176849403236236755819825, 4.13371225680253503070498848818, 4.97301981344827521216218300729, 5.34191187961267423810069103355, 7.17554049140179898393748176428, 8.039551381068635006336518359315, 8.746918059034118969648002242594, 9.170530683705467615700160592455, 10.09269433845270582164810546999

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