Properties

Label 2-1216-152.69-c0-0-0
Degree $2$
Conductor $1216$
Sign $-0.163 - 0.986i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.163 - 0.986i$
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: \((2,\ 1216,\ (\ :0),\ -0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340871427\)
\(L(\frac12)\) \(\approx\) \(1.340871427\)
\(L(1)\) 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

−9.859825295837904118138054437879, −9.391416534658203502892946091109, −8.922177035390530661016816947780, −7.73307144997249478749434301097, −7.15159434707151250042789265430, −5.63838339759436784657694854294, −4.80308151849610231064714743141, −4.14816371993926317542914870285, −3.14486198115345709746484606932, −2.21709606127198372665030073163, 1.16938147683957324175620717374, 2.33145767731444444691075459073, 3.20312060153324186331548047291, 4.34459692288072221695634179016, 6.07511311539990027229873017074, 6.24973198387071939405427140802, 7.46563667873310485974817351383, 7.989860403924489491363240210229, 8.725386007813381043532035671217, 9.335797461812900121233535170021

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