Properties

Label 2-1216-8.5-c1-0-16
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  i·3-s − 3·7-s + 2·9-s + 3i·13-s + 3·17-s + i·19-s + 3i·21-s + 9·23-s + 5·25-s − 5i·27-s − 9i·29-s − 6·31-s − 6i·37-s + 3·39-s + 6·41-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.13·7-s + 0.666·9-s + 0.832i·13-s + 0.727·17-s + 0.229i·19-s + 0.654i·21-s + 1.87·23-s + 25-s − 0.962i·27-s − 1.67i·29-s − 1.07·31-s − 0.986i·37-s + 0.480·39-s + 0.937·41-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: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.560044459\)
\(L(\frac12)\) \(\approx\) \(1.560044459\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 - 9T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 3iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.452082351369450189442281696885, −9.064455935931297151545476565667, −7.77307587699736122115391753624, −7.08070584991519485848520445672, −6.50250727412608230979435762145, −5.55270838919019823822788404583, −4.35336587019880043428892678064, −3.39567998387535460812683799868, −2.24773209460051432385704753352, −0.859629915268150574693685825194, 1.09513157184347448438907192422, 3.01000527337709967171374865417, 3.46764598695459551704989318974, 4.79260310657974610597233093823, 5.43690883218433792305781071075, 6.69205073326629599870617058586, 7.16052590256910063539794378756, 8.312700185860270809450382307879, 9.338297278609692927807365341382, 9.661165550558986193511889780429

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