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Properties

Label 2-6e2-36.7-c2-0-6
Degree $2$
Conductor $36$
Sign $0.764 + 0.645i$
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  + (0.710 − 1.86i)2-s + (2.32 + 1.89i)3-s + (−2.98 − 2.65i)4-s + (1.35 + 2.34i)5-s + (5.20 − 2.99i)6-s + (−10.0 − 5.79i)7-s + (−7.09 + 3.70i)8-s + (1.78 + 8.82i)9-s + (5.35 − 0.865i)10-s + (8.54 + 4.93i)11-s + (−1.89 − 11.8i)12-s + (0.296 + 0.513i)13-s + (−17.9 + 14.6i)14-s + (−1.31 + 8.03i)15-s + (1.87 + 15.8i)16-s − 8.87·17-s + ⋯
L(s)  = 1  + (0.355 − 0.934i)2-s + (0.774 + 0.633i)3-s + (−0.747 − 0.664i)4-s + (0.271 + 0.469i)5-s + (0.866 − 0.498i)6-s + (−1.43 − 0.828i)7-s + (−0.886 + 0.462i)8-s + (0.198 + 0.980i)9-s + (0.535 − 0.0865i)10-s + (0.777 + 0.448i)11-s + (−0.158 − 0.987i)12-s + (0.0227 + 0.0394i)13-s + (−1.28 + 1.04i)14-s + (−0.0873 + 0.535i)15-s + (0.117 + 0.993i)16-s − 0.522·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.645i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.764 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.764 + 0.645i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ 0.764 + 0.645i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22647 - 0.448519i\)
\(L(\frac12)\) \(\approx\) \(1.22647 - 0.448519i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.710 + 1.86i)T \)
3 \( 1 + (-2.32 - 1.89i)T \)
good5 \( 1 + (-1.35 - 2.34i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (10.0 + 5.79i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.54 - 4.93i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.296 - 0.513i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 8.87T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (-18.2 + 10.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.1 + 17.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (14.3 - 8.27i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 40.6T + 1.36e3T^{2} \)
41 \( 1 + (-21.2 - 36.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.2 - 18.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.57 + 0.907i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 21.1T + 2.80e3T^{2} \)
59 \( 1 + (76.6 - 44.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36.4 + 63.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38.3 + 22.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 + (-8.30 - 4.79i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (73.6 + 42.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 64.7T + 7.92e3T^{2} \)
97 \( 1 + (3.59 - 6.22i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.88123216881566661920202166318, −14.61931598621756258153504938102, −13.69165739074757757244417544857, −12.74676762892163646381613264180, −10.90608325443894486903958984362, −9.948844445594739325128681377008, −9.074479966503906656973916399668, −6.69527199257387943302473614736, −4.31121594421206831093562940485, −2.93214743671620789930773005350, 3.39813769776261327187346693904, 5.86868207773584810548859013079, 7.00665309684294756878561873834, 8.779681722144262847043701271722, 9.330635442117715583963071293020, 12.27398391594816195455273786390, 12.97212234484907880796190037680, 13.98952117041464156743898125286, 15.19939208643522957984737138612, 16.19321962979793826957682361703

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