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Properties

Label 2-1216-152.75-c1-0-0
Degree $2$
Conductor $1216$
Sign $-0.965 - 0.258i$
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  − 3.04i·5-s + 5.27i·7-s + 3·9-s − 6.50·11-s − 7.27·17-s − 4.35·19-s − 4i·23-s − 4.27·25-s + 16.0·35-s − 5.67·43-s − 9.13i·45-s − 2.72i·47-s − 20.8·49-s + 19.8i·55-s + 10.8i·61-s + ⋯
L(s)  = 1  − 1.36i·5-s + 1.99i·7-s + 9-s − 1.96·11-s − 1.76·17-s − 1.00·19-s − 0.834i·23-s − 0.854·25-s + 2.71·35-s − 0.865·43-s − 1.36i·45-s − 0.397i·47-s − 2.97·49-s + 2.67i·55-s + 1.38i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.965 - 0.258i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1335757901\)
\(L(\frac12)\) \(\approx\) \(0.1335757901\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 3.04iT - 5T^{2} \)
7 \( 1 - 5.27iT - 7T^{2} \)
11 \( 1 + 6.50T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 7.27T + 17T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 5.67T + 43T^{2} \)
47 \( 1 + 2.72iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10.8iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 5.82T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.71T + 83T^{2} \)
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

−10.01301001947091259370450403901, −8.995410528875449701325478033096, −8.627406756372271797291653003942, −7.991801306603278930125788226282, −6.69348735058433647175147127318, −5.71499435825021164142733899242, −4.96255007415563520042436330131, −4.44679112211103057178895879795, −2.61257758447966562836728982884, −1.95791676164844489658401001448, 0.05137820674468295480397290008, 1.96166224219467299219958504918, 3.12142272396226625215652180262, 4.11936313809395067975247173826, 4.82911573682017179764690877900, 6.35068932084957571124918586841, 7.05136585208376924090542535745, 7.43887073941199374416263472190, 8.268078268454776369000080264533, 9.804577339978728741057540656998

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