Properties

Label 2-1216-19.11-c1-0-28
Degree $2$
Conductor $1216$
Sign $0.986 + 0.165i$
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  + (−0.866 + 1.5i)3-s + (2.09 − 3.63i)5-s + 4.28·7-s + 4.71·11-s + (−0.615 − 1.06i)13-s + (3.63 + 6.29i)15-s + (−1.61 + 2.79i)17-s + (−0.625 − 4.31i)19-s + (−3.71 + 6.43i)21-s + (1.90 + 3.29i)23-s + (−6.31 − 10.9i)25-s − 5.19·27-s + (−2.09 − 3.63i)29-s + 0.825·31-s + (−4.08 + 7.07i)33-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.938 − 1.62i)5-s + 1.62·7-s + 1.42·11-s + (−0.170 − 0.295i)13-s + (0.938 + 1.62i)15-s + (−0.391 + 0.678i)17-s + (−0.143 − 0.989i)19-s + (−0.810 + 1.40i)21-s + (0.397 + 0.687i)23-s + (−1.26 − 2.18i)25-s − 1.00·27-s + (−0.389 − 0.675i)29-s + 0.148·31-s + (−0.710 + 1.23i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.986 + 0.165i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.986 + 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.127743415\)
\(L(\frac12)\) \(\approx\) \(2.127743415\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.625 + 4.31i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.09 + 3.63i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.28T + 7T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 + (0.615 + 1.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.61 - 2.79i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.90 - 3.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.09 + 3.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.825T + 31T^{2} \)
37 \( 1 + 2.96T + 37T^{2} \)
41 \( 1 + (-2.69 + 4.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.49 - 2.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.653 + 1.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.38 + 2.39i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.31 - 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.09 - 7.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.69 - 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.04 - 7.01i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.69 - 8.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.31 + 4.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.32T + 83T^{2} \)
89 \( 1 + (5.61 + 9.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.92 - 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.635625732743177770997310910879, −8.900637335652653662485985108021, −8.467808294542287608557160017106, −7.32101097930249377759373064582, −5.91540048374840047871651602628, −5.35064817680255238524958022967, −4.53417916606994304312776649660, −4.20446982534396736646300732814, −1.98088980597585367668377714937, −1.18128176413255367918321264832, 1.48033238974674567267175112715, 2.06169113165797556832147223520, 3.47064726783815038385657704151, 4.72813100263886450900511061483, 5.86452808009993485213285628680, 6.54417106974782793043651361216, 7.05397920702501755410134905023, 7.83551857615272581042419834344, 9.013457399482249135790495626722, 9.807481959652829483841254168917

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