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Properties

Label 2-6e2-36.7-c2-0-4
Degree $2$
Conductor $36$
Sign $0.939 - 0.342i$
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  + 2·2-s + (−1.5 + 2.59i)3-s + 4·4-s + (−2 − 3.46i)5-s + (−3 + 5.19i)6-s + (−3 − 1.73i)7-s + 8·8-s + (−4.5 − 7.79i)9-s + (−4 − 6.92i)10-s + (−10.5 − 6.06i)11-s + (−6 + 10.3i)12-s + (11 + 19.0i)13-s + (−6 − 3.46i)14-s + 12·15-s + 16·16-s − 11·17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.400 − 0.692i)5-s + (−0.5 + 0.866i)6-s + (−0.428 − 0.247i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.400 − 0.692i)10-s + (−0.954 − 0.551i)11-s + (−0.5 + 0.866i)12-s + (0.846 + 1.46i)13-s + (−0.428 − 0.247i)14-s + 0.800·15-s + 16-s − 0.647·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.980928\)
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: \((2,\ 36,\ (\ :1),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.39495 + 0.245968i\)
\(L(\frac12)\) \(\approx\) \(1.39495 + 0.245968i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (1.5 - 2.59i)T \)
good5 \( 1 + (2 + 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3 + 1.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.5 + 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11 - 19.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 11T + 289T^{2} \)
19 \( 1 - 15.5iT - 361T^{2} \)
23 \( 1 + (-21 + 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (17 - 29.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-6 + 3.46i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + (6.5 + 11.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (43.5 + 25.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-3 - 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 52T + 2.80e3T^{2} \)
59 \( 1 + (46.5 - 26.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-100.5 + 58.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + (-24 - 13.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-30 - 17.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 2T + 7.92e3T^{2} \)
97 \( 1 + (-21.5 + 37.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−16.34747080257749978354338129891, −15.31152830152439919587319880642, −13.91108130715543254597293570290, −12.70593532802523966944013520067, −11.50322480831859729972002717860, −10.51376092603428717759311978896, −8.716466896375390396784789917774, −6.56631625612521158261131161600, −5.05569964022903767646520307504, −3.75814638097851894193692459702, 2.89271783766293095826803549691, 5.34152452828346158597299259661, 6.70852912794211773634639399414, 7.83199852382580028106287117428, 10.60879141018227534984727405555, 11.45179474174408047335836239829, 12.91752311088903747498976557362, 13.33597851826392897848034759447, 15.12368746187095454049426872587, 15.74632055140607307785414647691

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