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Properties

Label 2-6e2-9.7-c1-0-0
Degree $2$
Conductor $36$
Sign $0.939 - 0.342i$
Analytic cond. $0.287461$
Root an. cond. $0.536154$
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  + 1.73i·3-s + (−1.5 − 2.59i)5-s + (0.5 − 0.866i)7-s − 2.99·9-s + (−1.5 + 2.59i)11-s + (0.5 + 0.866i)13-s + (4.5 − 2.59i)15-s + 6·17-s − 4·19-s + (1.49 + 0.866i)21-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s − 5.19i·27-s + (−1.5 + 2.59i)29-s + (−2.5 − 4.33i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.670 − 1.16i)5-s + (0.188 − 0.327i)7-s − 0.999·9-s + (−0.452 + 0.783i)11-s + (0.138 + 0.240i)13-s + (1.16 − 0.670i)15-s + 1.45·17-s − 0.917·19-s + (0.327 + 0.188i)21-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.999i·27-s + (−0.278 + 0.482i)29-s + (−0.449 − 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685176 + 0.120815i\)
\(L(\frac12)\) \(\approx\) \(0.685176 + 0.120815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−16.53794698895738274335983513303, −15.56565942816637594749342768540, −14.51124490609267068181895238046, −12.87316257412045570170123041937, −11.72243564776574288919111744591, −10.32607613215945308685700334377, −9.052069465832944814813863569504, −7.80051601622674967563616824760, −5.27306770539016633810253170969, −4.04092755146920523894353866314, 3.03859835267122871133236527477, 5.93803940801848976957527409700, 7.34677056584845444896322601026, 8.403840569983277239534394947763, 10.61822029052000652311156728651, 11.61412770937499088820529268108, 12.81311426566911892789031311059, 14.20904714041741431979615853742, 15.02120898329480019417674318471, 16.52017147857990237778726022712

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