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Properties

Label 1-3616-3616.1155-r1-0-0
Degree $1$
Conductor $3616$
Sign $-0.902 + 0.430i$
Analytic cond. $388.593$
Root an. cond. $388.593$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.781 + 0.623i)3-s + (−0.781 + 0.623i)5-s + (0.781 + 0.623i)7-s + (0.222 + 0.974i)9-s + (0.943 + 0.330i)11-s + (0.111 + 0.993i)13-s − 15-s + (0.846 − 0.532i)17-s + (0.623 − 0.781i)19-s + (0.222 + 0.974i)21-s + (−0.993 − 0.111i)23-s + (0.222 − 0.974i)25-s + (−0.433 + 0.900i)27-s + (0.974 + 0.222i)29-s + (−0.781 − 0.623i)31-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)3-s + (−0.781 + 0.623i)5-s + (0.781 + 0.623i)7-s + (0.222 + 0.974i)9-s + (0.943 + 0.330i)11-s + (0.111 + 0.993i)13-s − 15-s + (0.846 − 0.532i)17-s + (0.623 − 0.781i)19-s + (0.222 + 0.974i)21-s + (−0.993 − 0.111i)23-s + (0.222 − 0.974i)25-s + (−0.433 + 0.900i)27-s + (0.974 + 0.222i)29-s + (−0.781 − 0.623i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3616 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3616 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3616\)    =    \(2^{5} \cdot 113\)
Sign: $-0.902 + 0.430i$
Analytic conductor: \(388.593\)
Root analytic conductor: \(388.593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3616} (1155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3616,\ (1:\ ),\ -0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7162883337 + 3.165663043i\)
\(L(\frac12)\) \(\approx\) \(0.7162883337 + 3.165663043i\)
\(L(1)\) \(\approx\) \(1.257647984 + 0.7607848262i\)
\(L(1)\) \(\approx\) \(1.257647984 + 0.7607848262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
113 \( 1 \)
good3 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (-0.781 + 0.623i)T \)
7 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (0.943 + 0.330i)T \)
13 \( 1 + (0.111 + 0.993i)T \)
17 \( 1 + (0.846 - 0.532i)T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (-0.993 - 0.111i)T \)
29 \( 1 + (0.974 + 0.222i)T \)
31 \( 1 + (-0.781 - 0.623i)T \)
37 \( 1 + (-0.433 - 0.900i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (0.943 - 0.330i)T \)
53 \( 1 + (0.943 - 0.330i)T \)
59 \( 1 + (0.781 + 0.623i)T \)
61 \( 1 + (-0.330 + 0.943i)T \)
67 \( 1 + (0.433 + 0.900i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.330 + 0.943i)T \)
83 \( 1 + (-0.846 + 0.532i)T \)
89 \( 1 + (-0.846 - 0.532i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.31074122371830810214882897130, −17.51703491845268832259607602486, −16.95161755805396295859161695729, −16.12873753859755013249293294321, −15.380137645809613953278697785316, −14.63887796404402453029784009032, −14.09279753457414326781631938553, −13.535317785735434763853944902295, −12.5104877358508760306071286100, −12.148982919123312036501187712259, −11.50053000496209761894445546642, −10.47412323927817639605536474332, −9.791180286713913681525481507878, −8.741395897277760989386656257001, −8.27019395479687563483350764968, −7.758773787237537752496690287946, −7.159314544229023456988151757141, −6.125416707723300909198544133644, −5.31835577697323017164792242495, −4.26397911443288490028083866132, −3.66722792180198027833361817247, −3.10697266161803667040514374973, −1.66507364560927156139495571688, −1.225752385121705557550672246464, −0.46060798072802957754301100630, 1.08175919695268547875411517804, 2.15368067925846397945036293458, 2.712010685217416909022573168898, 3.80553396101458001192258483588, 4.1418901528215304957075475515, 5.01256619309441141625555267281, 5.88921579799902317529212906383, 7.18485942027084721956953835340, 7.33280686227936919027098034116, 8.45533268422452990433321969064, 8.867017839660540370788182515950, 9.63793719805009534338949234502, 10.39097545284485370709773029665, 11.27691302823808853427764222825, 11.78219817597763865530691583849, 12.30130659532891488194576537297, 13.669635781760771345445657936865, 14.254969745014022835187125113030, 14.5861206136252545675516828906, 15.298526875248836959609449656431, 15.98590301155320986831547283963, 16.46105037781912684573040317558, 17.51232104316491595656277429680, 18.303611901677141231895668458715, 18.86140912185756181319081942235

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