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Properties

Label 2-6e2-9.2-c4-0-3
Degree $2$
Conductor $36$
Sign $-0.970 - 0.239i$
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  + (−4.23 − 7.94i)3-s + (−34.8 + 20.1i)5-s + (−7.38 + 12.7i)7-s + (−45.1 + 67.2i)9-s + (−70.7 − 40.8i)11-s + (−139. − 240. i)13-s + (307. + 191. i)15-s + 10.8i·17-s + 532.·19-s + (132. + 4.45i)21-s + (−702. + 405. i)23-s + (498. − 862. i)25-s + (725. + 73.0i)27-s + (−257. − 148. i)29-s + (−97.5 − 168. i)31-s + ⋯
L(s)  = 1  + (−0.470 − 0.882i)3-s + (−1.39 + 0.805i)5-s + (−0.150 + 0.261i)7-s + (−0.556 + 0.830i)9-s + (−0.584 − 0.337i)11-s + (−0.822 − 1.42i)13-s + (1.36 + 0.851i)15-s + 0.0376i·17-s + 1.47·19-s + (0.301 + 0.0100i)21-s + (−1.32 + 0.766i)23-s + (0.797 − 1.38i)25-s + (0.994 + 0.100i)27-s + (−0.306 − 0.176i)29-s + (−0.101 − 0.175i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.970 - 0.239i$
Analytic conductor: \(3.72131\)
Root analytic conductor: \(1.92907\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :2),\ -0.970 - 0.239i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00736153 + 0.0606731i\)
\(L(\frac12)\) \(\approx\) \(0.00736153 + 0.0606731i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.23 + 7.94i)T \)
good5 \( 1 + (34.8 - 20.1i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (7.38 - 12.7i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (70.7 + 40.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (139. + 240. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 10.8iT - 8.35e4T^{2} \)
19 \( 1 - 532.T + 1.30e5T^{2} \)
23 \( 1 + (702. - 405. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (257. + 148. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (97.5 + 168. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 2.09e3T + 1.87e6T^{2} \)
41 \( 1 + (1.35e3 - 784. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-46.0 + 79.8i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.84e3 - 1.06e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.57e3iT - 7.89e6T^{2} \)
59 \( 1 + (-1.34e3 + 778. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.68e3 - 4.65e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (457. + 792. i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 8.21e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.43e3T + 2.83e7T^{2} \)
79 \( 1 + (2.31e3 - 4.01e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (5.19e3 + 2.99e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 8.43e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.01e3 + 5.22e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.22990890221794434048508546200, −13.79054205064172113865550016246, −12.36505304407082905131579986634, −11.59776853148123733746320674098, −10.37676583117061858744253504742, −7.978411545414513420961573097287, −7.34046649286107264610101171890, −5.54632005068896516625825285142, −3.08850342555356189597595059128, −0.04346651088423851178572014125, 3.92454018233143933303849856510, 5.03307722637314383686000020940, 7.27537431848903114295284106343, 8.824825128277255640254231263083, 10.13789369601212504809775215142, 11.68781520739039609302456240061, 12.23313051891864589926368395967, 14.17116481257107056246664887384, 15.58426531897205515512103976525, 16.18083905215443605297021899058

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