Properties

Label 2-1216-152.75-c1-0-23
Degree $2$
Conductor $1216$
Sign $0.621 + 0.783i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.621 + 0.783i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.820889175\)
\(L(\frac12)\) \(\approx\) \(1.820889175\)
\(L(\frac{3}{2})\) 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

−9.818976983066798320017690413751, −8.657406056226757029392131663584, −7.68191606790161535292344126828, −7.09704410691534492595996752076, −6.58615030010862309697059409641, −5.66451625682008719978681713578, −4.29623557514844464208380390629, −3.30091327831325657875194798288, −2.09745859622874046655580551569, −1.01102998948017035303886863168, 1.17134323649567858376662904904, 2.81120755735681013721697386347, 3.95611627419880782726378646121, 4.68759101457801892524595668614, 5.31007649048149705749361548648, 6.44748727415321766850422115321, 7.41933797707272634345819792798, 8.524737632856454398447781502932, 9.349733206506728448831985105173, 9.544681384845321449699929084047

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