Properties

Label 2-1216-19.18-c2-0-0
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·3-s − 7·5-s − 11·7-s − 23.0·9-s + 3·11-s + 11.3i·13-s − 39.5i·15-s − 17·17-s − 19·19-s − 62.2i·21-s − 2·23-s + 24·25-s − 79.1i·27-s + 39.5i·29-s − 5.65i·31-s + ⋯
L(s)  = 1  + 1.88i·3-s − 1.40·5-s − 1.57·7-s − 2.55·9-s + 0.272·11-s + 0.870i·13-s − 2.63i·15-s − 17-s − 19-s − 2.96i·21-s − 0.0869·23-s + 0.959·25-s − 2.93i·27-s + 1.36i·29-s − 0.182i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $1$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06792195590\)
\(L(\frac12)\) \(\approx\) \(0.06792195590\)
\(L(2)\) 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 \( 1 + 19T \)
good3 \( 1 - 5.65iT - 9T^{2} \)
5 \( 1 + 7T + 25T^{2} \)
7 \( 1 + 11T + 49T^{2} \)
11 \( 1 - 3T + 121T^{2} \)
13 \( 1 - 11.3iT - 169T^{2} \)
17 \( 1 + 17T + 289T^{2} \)
23 \( 1 + 2T + 529T^{2} \)
29 \( 1 - 39.5iT - 841T^{2} \)
31 \( 1 + 5.65iT - 961T^{2} \)
37 \( 1 - 39.5iT - 1.36e3T^{2} \)
41 \( 1 - 39.5iT - 1.68e3T^{2} \)
43 \( 1 + 21T + 1.84e3T^{2} \)
47 \( 1 - 5T + 2.20e3T^{2} \)
53 \( 1 + 5.65iT - 2.80e3T^{2} \)
59 \( 1 - 33.9iT - 3.48e3T^{2} \)
61 \( 1 + 23T + 3.72e3T^{2} \)
67 \( 1 - 39.5iT - 4.48e3T^{2} \)
71 \( 1 + 90.5iT - 5.04e3T^{2} \)
73 \( 1 - 39T + 5.32e3T^{2} \)
79 \( 1 - 96.1iT - 6.24e3T^{2} \)
83 \( 1 + 6T + 6.88e3T^{2} \)
89 \( 1 - 118. iT - 7.92e3T^{2} \)
97 \( 1 + 169. iT - 9.40e3T^{2} \)
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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

−10.31150705842521774838538054939, −9.499504841946765614837880419047, −8.938076511698764801202415780583, −8.277787125946209191810643769546, −6.87427926301022186856762226721, −6.24383336522533258810506020173, −4.89053807554149028884187376236, −4.16865018399797391887960700234, −3.64239698037817293357079682601, −2.82158798408060234573502600703, 0.04288121501771759171089913368, 0.48773837777327157574429027387, 2.21794971763085243798823570837, 3.15777091138946962357321195091, 4.08562084291334047450648913169, 5.74116235583849464221598370865, 6.50506623033683166689465842016, 7.02308606120827680893592290552, 7.80743403561272948359982110156, 8.431825644146934926656598884659

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