Properties

Label 2-6e2-12.11-c15-0-25
Degree $2$
Conductor $36$
Sign $-0.975 - 0.221i$
Analytic cond. $51.3696$
Root an. cond. $7.16726$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (100. − 150. i)2-s + (−1.25e4 − 3.02e4i)4-s + 2.54e5i·5-s + 1.64e6i·7-s + (−5.81e6 − 1.16e6i)8-s + (3.82e7 + 2.55e7i)10-s + 1.11e8·11-s − 3.82e8·13-s + (2.48e8 + 1.65e8i)14-s + (−7.60e8 + 7.57e8i)16-s − 1.56e9i·17-s − 2.90e9i·19-s + (7.69e9 − 3.17e9i)20-s + (1.12e10 − 1.67e10i)22-s − 3.02e10·23-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)2-s + (−0.381 − 0.924i)4-s + 1.45i·5-s + 0.756i·7-s + (−0.980 − 0.196i)8-s + (1.20 + 0.808i)10-s + 1.72·11-s − 1.69·13-s + (0.628 + 0.420i)14-s + (−0.708 + 0.705i)16-s − 0.926i·17-s − 0.746i·19-s + (1.34 − 0.555i)20-s + (0.957 − 1.43i)22-s − 1.85·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.975 - 0.221i$
Analytic conductor: \(51.3696\)
Root analytic conductor: \(7.16726\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :15/2),\ -0.975 - 0.221i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.3205819874\)
\(L(\frac12)\) \(\approx\) \(0.3205819874\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-100. + 150. i)T \)
3 \( 1 \)
good5 \( 1 - 2.54e5iT - 3.05e10T^{2} \)
7 \( 1 - 1.64e6iT - 4.74e12T^{2} \)
11 \( 1 - 1.11e8T + 4.17e15T^{2} \)
13 \( 1 + 3.82e8T + 5.11e16T^{2} \)
17 \( 1 + 1.56e9iT - 2.86e18T^{2} \)
19 \( 1 + 2.90e9iT - 1.51e19T^{2} \)
23 \( 1 + 3.02e10T + 2.66e20T^{2} \)
29 \( 1 + 1.00e11iT - 8.62e21T^{2} \)
31 \( 1 + 5.87e9iT - 2.34e22T^{2} \)
37 \( 1 + 1.44e11T + 3.33e23T^{2} \)
41 \( 1 - 6.03e11iT - 1.55e24T^{2} \)
43 \( 1 + 2.24e12iT - 3.17e24T^{2} \)
47 \( 1 - 5.07e10T + 1.20e25T^{2} \)
53 \( 1 - 1.49e12iT - 7.31e25T^{2} \)
59 \( 1 + 1.98e13T + 3.65e26T^{2} \)
61 \( 1 + 1.36e13T + 6.02e26T^{2} \)
67 \( 1 - 1.82e13iT - 2.46e27T^{2} \)
71 \( 1 + 1.20e14T + 5.87e27T^{2} \)
73 \( 1 + 5.71e13T + 8.90e27T^{2} \)
79 \( 1 - 8.13e11iT - 2.91e28T^{2} \)
83 \( 1 + 3.10e14T + 6.11e28T^{2} \)
89 \( 1 + 4.39e14iT - 1.74e29T^{2} \)
97 \( 1 + 4.22e13T + 6.33e29T^{2} \)
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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

−12.01628450885486663492093434744, −11.66205214536543438964079356059, −10.15527961401446572476336355729, −9.274275763676345117684651389794, −7.08039961172908558504741101335, −5.96168974838181559846231329793, −4.31490853703021687061368402214, −2.91213150784157386541318339895, −2.05800673364799525785453663681, −0.06621912134684032588502046437, 1.49739329272349995320507029595, 3.87923172063537214939134638402, 4.64721719705279197662152306138, 6.04950308163240696639409504375, 7.45219640900021204546729718137, 8.631070927755920187042353600843, 9.756734790735316303993725814882, 12.06280208900615581516571507253, 12.52952016999747986634413880986, 13.95918720305118417143606394055

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