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Properties

Label 2-6e2-9.2-c2-0-1
Degree $2$
Conductor $36$
Sign $0.724 + 0.689i$
Analytic cond. $0.980928$
Root an. cond. $0.990418$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.686 − 2.92i)3-s + (6.55 − 3.78i)5-s + (−4.55 + 7.89i)7-s + (−8.05 + 4.00i)9-s + (−0.383 − 0.221i)11-s + (5.55 + 9.62i)13-s + (−15.5 − 16.5i)15-s + 8.01i·17-s − 8.11·19-s + (26.1 + 7.89i)21-s + (20.4 − 11.8i)23-s + (16.1 − 28.0i)25-s + (17.2 + 20.7i)27-s + (−45.9 − 26.5i)29-s + (−14.6 − 25.4i)31-s + ⋯
L(s)  = 1  + (−0.228 − 0.973i)3-s + (1.31 − 0.757i)5-s + (−0.651 + 1.12i)7-s + (−0.895 + 0.445i)9-s + (−0.0348 − 0.0201i)11-s + (0.427 + 0.740i)13-s + (−1.03 − 1.10i)15-s + 0.471i·17-s − 0.427·19-s + (1.24 + 0.375i)21-s + (0.888 − 0.513i)23-s + (0.647 − 1.12i)25-s + (0.638 + 0.769i)27-s + (−1.58 − 0.913i)29-s + (−0.473 − 0.819i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ 0.724 + 0.689i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.986472 - 0.394134i\)
\(L(\frac12)\) \(\approx\) \(0.986472 - 0.394134i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.686 + 2.92i)T \)
good5 \( 1 + (-6.55 + 3.78i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (4.55 - 7.89i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (0.383 + 0.221i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.55 - 9.62i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 8.01iT - 289T^{2} \)
19 \( 1 + 8.11T + 361T^{2} \)
23 \( 1 + (-20.4 + 11.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (45.9 + 26.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.6 + 25.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 18.4T + 1.36e3T^{2} \)
41 \( 1 + (38.9 - 22.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (11.5 - 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (7.32 + 4.22i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 60.5iT - 2.80e3T^{2} \)
59 \( 1 + (-65.9 + 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.67 - 4.63i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-54.8 - 95.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35T + 5.32e3T^{2} \)
79 \( 1 + (0.792 - 1.37i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (7.32 + 4.22i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (57.6 - 99.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−16.48028443867104973548909204840, −14.75901125965441892249716022187, −13.23864445292210477320653382431, −12.87615700747883386573401816758, −11.47694210921832043824015101218, −9.555882657032207902577351507934, −8.564714299984231542017045099162, −6.45509526001152787510918040999, −5.54298477534478364165456106683, −2.07470438107883879783765653509, 3.37995410589960562088165736252, 5.50393391450211072285070926941, 6.90344025080803101988364865456, 9.248185180654348138590714608439, 10.28505753112685403611954516339, 10.93161879084923963863614374360, 13.09736514024626647317515534864, 14.06521141021762032981694563471, 15.18634692253273678256855053114, 16.56309715794775070019642513502

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