Properties

Label 2-6e2-36.31-c4-0-6
Degree $2$
Conductor $36$
Sign $0.367 - 0.930i$
Analytic cond. $3.72131$
Root an. cond. $1.92907$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 − 3.54i)2-s + (6.53 + 6.18i)3-s + (−9.11 + 13.1i)4-s + (−14.8 + 25.7i)5-s + (9.78 − 34.6i)6-s + (−51.8 + 29.9i)7-s + (63.5 + 7.86i)8-s + (4.46 + 80.8i)9-s + (118. + 4.89i)10-s + (195. − 112. i)11-s + (−140. + 29.6i)12-s + (−85.8 + 148. i)13-s + (202. + 128. i)14-s + (−256. + 76.4i)15-s + (−90.0 − 239. i)16-s − 99.0·17-s + ⋯
L(s)  = 1  + (−0.464 − 0.885i)2-s + (0.726 + 0.687i)3-s + (−0.569 + 0.822i)4-s + (−0.595 + 1.03i)5-s + (0.271 − 0.962i)6-s + (−1.05 + 0.611i)7-s + (0.992 + 0.122i)8-s + (0.0551 + 0.998i)9-s + (1.18 + 0.0489i)10-s + (1.61 − 0.931i)11-s + (−0.978 + 0.205i)12-s + (−0.508 + 0.880i)13-s + (1.03 + 0.654i)14-s + (−1.14 + 0.339i)15-s + (−0.351 − 0.936i)16-s − 0.342·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(3.72131\)
Root analytic conductor: \(1.92907\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :2),\ 0.367 - 0.930i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.864686 + 0.588208i\)
\(L(\frac12)\) \(\approx\) \(0.864686 + 0.588208i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.85 + 3.54i)T \)
3 \( 1 + (-6.53 - 6.18i)T \)
good5 \( 1 + (14.8 - 25.7i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (51.8 - 29.9i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-195. + 112. i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (85.8 - 148. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 99.0T + 8.35e4T^{2} \)
19 \( 1 + 169. iT - 1.30e5T^{2} \)
23 \( 1 + (-310. - 179. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-9.01 - 15.6i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-671. - 387. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 609.T + 1.87e6T^{2} \)
41 \( 1 + (-206. + 357. i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-265. + 153. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-2.27e3 + 1.31e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.03e3T + 7.89e6T^{2} \)
59 \( 1 + (2.25e3 + 1.29e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-708. - 1.22e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-5.19e3 - 2.99e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.23e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.06e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.63e3 + 944. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (5.83e3 - 3.36e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 9.43e3T + 6.27e7T^{2} \)
97 \( 1 + (-7.29e3 - 1.26e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.95611778200246311700713875750, −14.67182885742483944112756706785, −13.60724006216600354494516606991, −11.91546085284032794969508992606, −10.96087158943875622571537259980, −9.539632247472633808776290922927, −8.795417456927492524264132321481, −6.93292878584391085596607715054, −3.91701965788238434163069170381, −2.83555144083086472289177694763, 0.837272819582801971131818610122, 4.19594542104458365811755036989, 6.51805446023412740443483657638, 7.60501221668009106456171740614, 8.893363793434215645112656367449, 9.821345880967125017888451703198, 12.27850752099664731843469804111, 13.15417963410401676729536895453, 14.47658795166194130429708762088, 15.49623922691047156298889167997

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