Properties

Label 2-1216-152.75-c1-0-18
Degree $2$
Conductor $1216$
Sign $0.707 + 0.707i$
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  − 2.27i·3-s + 4.53i·7-s − 2.16·9-s − 6.13·13-s + 8.23·17-s − 4.35i·19-s + 10.3·21-s − 2.86i·23-s + 5·25-s − 1.90i·27-s + 10.6·29-s + 8.71·37-s + 13.9i·39-s − 6i·47-s − 13.5·49-s + ⋯
L(s)  = 1  − 1.31i·3-s + 1.71i·7-s − 0.721·9-s − 1.70·13-s + 1.99·17-s − 0.999i·19-s + 2.24·21-s − 0.596i·23-s + 25-s − 0.365i·27-s + 1.98·29-s + 1.43·37-s + 2.23i·39-s − 0.875i·47-s − 1.93·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.707 + 0.707i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.646915236\)
\(L(\frac12)\) \(\approx\) \(1.646915236\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35iT \)
good3 \( 1 + 2.27iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 4.53iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6.13T + 13T^{2} \)
17 \( 1 - 8.23T + 17T^{2} \)
23 \( 1 + 2.86iT - 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 2.95T + 53T^{2} \)
59 \( 1 - 14.5iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 5.44iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.550657184560204757759654269107, −8.637423423320998626023004055252, −7.960405021139775983946064144449, −7.15732572974214134239064446544, −6.40339508809444720153832924106, −5.49276929665662341063713062184, −4.77682314229197190075553929930, −2.73827443079808372802454640087, −2.51054089492660453812627288728, −0.963458208018100116838881629643, 1.02530149836196833590649164211, 3.01055252533244251913754319744, 3.79995782820614073504658218266, 4.65565593395405416654858457734, 5.22757840029787960855859941329, 6.56444340437325956396901072867, 7.56858719772299876678717381202, 8.009119422180190036800312882315, 9.504087893590616468756249506970, 9.954129244305278410782361784373

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