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Properties

Label 2-1216-8.5-c1-0-3
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  + 2i·3-s + 1.73i·5-s − 1.73·7-s − 9-s + 3i·11-s − 3.46·15-s − 3·17-s + i·19-s − 3.46i·21-s − 3.46·23-s + 2.00·25-s + 4i·27-s + 3.46i·29-s − 6.92·31-s − 6·33-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.774i·5-s − 0.654·7-s − 0.333·9-s + 0.904i·11-s − 0.894·15-s − 0.727·17-s + 0.229i·19-s − 0.755i·21-s − 0.722·23-s + 0.400·25-s + 0.769i·27-s + 0.643i·29-s − 1.24·31-s − 1.04·33-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: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9348573010\)
\(L(\frac12)\) \(\approx\) \(0.9348573010\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 5.19iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−10.19796513923060038371380488292, −9.457772776547212764860345878719, −8.914599592857417346400844064429, −7.55505285330425542879949809730, −6.90739691683867808223964668136, −5.97740398979658914658892958820, −4.91954337306519218008702268791, −4.04664179287802178979533292477, −3.30172783207716918971575777861, −2.10934568023208294799193817828, 0.38882321693785586860652856629, 1.58264610748522764387088759791, 2.79525138281833089959285908556, 4.01092049735730250287419843736, 5.13484712119125654347457659588, 6.20004686601810393127131810002, 6.67083769185318224221616767988, 7.71583901145264886938502256794, 8.381930178982780342453711027945, 9.142736831485046714372419527165

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