Properties

Label 2-1216-152.69-c0-0-1
Degree $2$
Conductor $1216$
Sign $-0.986 + 0.163i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $0$
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 + (−1 + 1.73i)9-s i·11-s + (−1 − 1.73i)17-s + (−0.866 − 0.5i)19-s + (−0.5 + 0.866i)25-s + 1.73·27-s + (−1.5 + 0.866i)33-s + (1.5 − 0.866i)41-s + (−1.73 + i)43-s − 49-s + (−1.73 + 3i)51-s + 1.73i·57-s + (0.866 + 1.5i)59-s + (0.866 − 1.5i)67-s + ⋯
L(s)  = 1  + (−0.866 − 1.5i)3-s + (−1 + 1.73i)9-s i·11-s + (−1 − 1.73i)17-s + (−0.866 − 0.5i)19-s + (−0.5 + 0.866i)25-s + 1.73·27-s + (−1.5 + 0.866i)33-s + (1.5 − 0.866i)41-s + (−1.73 + i)43-s − 49-s + (−1.73 + 3i)51-s + 1.73i·57-s + (0.866 + 1.5i)59-s + (0.866 − 1.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.986 + 0.163i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ -0.986 + 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5333147518\)
\(L(\frac12)\) \(\approx\) \(0.5333147518\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.404338275434048463896808663126, −8.590707903987930337157450131496, −7.68684135028316677525182091881, −6.97711602030787978597865787828, −6.33257708674911452047802337702, −5.53776359687503644312219280328, −4.63205338091576310812121474394, −3.00488165892666332850729556221, −1.92954769839779621893925985129, −0.49780959635767278189143083260, 2.09417791663724743545999534168, 3.77572869709346389763175668568, 4.27618592708093664970233098100, 5.08077694003217113156156636332, 6.10045168952924732326500265181, 6.64648105408904582220075486021, 8.108389503292234681425570323330, 8.823716397578462684019317027308, 9.909127900551651947405969471774, 10.14847671977434335862292516836

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