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Properties

Label 2-1216-152.125-c1-0-18
Degree $2$
Conductor $1216$
Sign $0.334 - 0.942i$
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  + (2.98 + 1.72i)3-s + (4.44 + 7.70i)9-s + 0.550i·11-s + (3 − 5.19i)17-s + (2.98 + 3.17i)19-s + (−2.5 − 4.33i)25-s + 20.3i·27-s + (−0.949 + 1.64i)33-s + (−6.39 + 11.0i)41-s + (−8.66 − 5i)43-s − 7·49-s + (17.9 − 10.3i)51-s + (3.44 + 14.6i)57-s + (8.00 + 4.62i)59-s + (12.4 − 7.17i)67-s + ⋯
L(s)  = 1  + (1.72 + 0.995i)3-s + (1.48 + 2.56i)9-s + 0.165i·11-s + (0.727 − 1.26i)17-s + (0.685 + 0.728i)19-s + (−0.5 − 0.866i)25-s + 3.91i·27-s + (−0.165 + 0.286i)33-s + (−0.999 + 1.73i)41-s + (−1.32 − 0.762i)43-s − 49-s + (2.50 − 1.44i)51-s + (0.456 + 1.93i)57-s + (1.04 + 0.601i)59-s + (1.51 − 0.876i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.334 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.075091303\)
\(L(\frac12)\) \(\approx\) \(3.075091303\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.98 - 3.17i)T \)
good3 \( 1 + (-2.98 - 1.72i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 0.550iT - 11T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (6.39 - 11.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.66 + 5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.00 - 4.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.4 + 7.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.84 + 13.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.55iT - 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.84 - 8.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.900786828032928979624687457870, −9.187355083675510626642811503619, −8.217716921084629177768957185828, −7.84082537024721209310678723887, −6.85515111481872845556310184759, −5.30572651147428362661828652947, −4.59942981828465454342946995849, −3.55259535391697080685741596785, −2.94030145689785725046673071303, −1.79805201381189175803210083353, 1.22129736066798647178126458339, 2.19177559420183175303914705057, 3.30329491063332326092019091470, 3.85999469430684076935262763719, 5.42888033495854709971728928865, 6.63885488060001536893507292017, 7.18904478356888911738273277285, 8.179800661520379619148474468354, 8.434273514363066760839695999947, 9.493135957189962539847921013607

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