Properties

Label 2-1216-152.75-c1-0-21
Degree $2$
Conductor $1216$
Sign $0.146 - 0.989i$
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  + 2i·3-s + 1.73i·5-s i·7-s − 9-s + 5.19·11-s + 4·13-s − 3.46·15-s + 3·17-s + (1.73 − 4i)19-s + 2·21-s + 2.00·25-s + 4i·27-s + 6·29-s − 10.3·31-s + 10.3i·33-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.774i·5-s − 0.377i·7-s − 0.333·9-s + 1.56·11-s + 1.10·13-s − 0.894·15-s + 0.727·17-s + (0.397 − 0.917i)19-s + 0.436·21-s + 0.400·25-s + 0.769i·27-s + 1.11·29-s − 1.86·31-s + 1.80i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.004714074\)
\(L(\frac12)\) \(\approx\) \(2.004714074\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 4i)T \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 5.19T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 5.19T + 43T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−9.997294797428401400488169751927, −9.167983310786597686778772271456, −8.615001603846070718651373667256, −7.22910173381065667945289459892, −6.69127609520359241104221571575, −5.66762329119144006741507770395, −4.58067596317813162544530753290, −3.73093099926058045251888252420, −3.18388995059805578943054428431, −1.34973367956283087424626317560, 1.14145275964948875984146320967, 1.64470188708365613517861056945, 3.32351565617078422486380381689, 4.29152722655187238719740439526, 5.57804828508678475203475401770, 6.24415098127566892862887630842, 7.04662657994024753516024805834, 7.932078467666181946483948772795, 8.722698568724154353646717962981, 9.245683521547658268288999368990

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