Properties

Label 2-1216-152.45-c1-0-32
Degree $2$
Conductor $1216$
Sign $0.163 + 0.986i$
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  + (2.59 − 1.5i)3-s + (−2.73 + 1.58i)5-s + 3.16·7-s + (3 − 5.19i)9-s − 3i·11-s + (−5.47 − 3.16i)13-s + (−4.74 + 8.21i)15-s + (−2 − 3.46i)17-s + (2.59 + 3.5i)19-s + (8.21 − 4.74i)21-s + (4.74 − 8.21i)23-s + (2.5 − 4.33i)25-s − 9i·27-s + (2.73 + 1.58i)29-s + 3.16·31-s + ⋯
L(s)  = 1  + (1.49 − 0.866i)3-s + (−1.22 + 0.707i)5-s + 1.19·7-s + (1 − 1.73i)9-s − 0.904i·11-s + (−1.51 − 0.877i)13-s + (−1.22 + 2.12i)15-s + (−0.485 − 0.840i)17-s + (0.596 + 0.802i)19-s + (1.79 − 1.03i)21-s + (0.989 − 1.71i)23-s + (0.5 − 0.866i)25-s − 1.73i·27-s + (0.508 + 0.293i)29-s + 0.567·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.163 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.337250625\)
\(L(\frac12)\) \(\approx\) \(2.337250625\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 3.5i)T \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.73 - 1.58i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + (5.47 + 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.74 + 8.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.73 - 1.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.16T + 31T^{2} \)
37 \( 1 - 3.16iT - 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.66 + 5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.58 + 2.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.06 - 3.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.73 - 1.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.16 - 5.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7iT - 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.248755629588511159811649172466, −8.324640287592510676063443013566, −7.999343297877788802965656511278, −7.35544856108590087561448808660, −6.73142991143023742672889384202, −5.12385000147357753421881776802, −4.10904834142080680208218199135, −2.92920177501395108500403244880, −2.58502255568730084760684048240, −0.875380605229083321842357486432, 1.71447707169420221586987797073, 2.79096046752039208001417702146, 4.05671917098784115592497634714, 4.55202058702493155760700008789, 5.03902867565527658910091266993, 7.22503380251550294986561629616, 7.65339421944893556212432471663, 8.271088214653250784285241025914, 9.276353058990302082405795416534, 9.397698917446058977570180955562

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