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Properties

Label 2-108-4.3-c4-0-0
Degree $2$
Conductor $108$
Sign $-0.639 + 0.768i$
Analytic cond. $11.1639$
Root an. cond. $3.34125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 + 3.76i)2-s + (−12.3 − 10.2i)4-s + 2.66·5-s + 88.8i·7-s + (55.2 − 32.3i)8-s + (−3.61 + 10.0i)10-s − 72.9i·11-s − 173.·13-s + (−334. − 120. i)14-s + (46.7 + 251. i)16-s − 423.·17-s − 153. i·19-s + (−32.7 − 27.2i)20-s + (274. + 99.1i)22-s − 723. i·23-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)2-s + (−0.768 − 0.639i)4-s + 0.106·5-s + 1.81i·7-s + (0.862 − 0.505i)8-s + (−0.0361 + 0.100i)10-s − 0.602i·11-s − 1.02·13-s + (−1.70 − 0.616i)14-s + (0.182 + 0.983i)16-s − 1.46·17-s − 0.424i·19-s + (−0.0818 − 0.0680i)20-s + (0.567 + 0.204i)22-s − 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.639 + 0.768i$
Analytic conductor: \(11.1639\)
Root analytic conductor: \(3.34125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :2),\ -0.639 + 0.768i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.127482 - 0.271787i\)
\(L(\frac12)\) \(\approx\) \(0.127482 - 0.271787i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 3.76i)T \)
3 \( 1 \)
good5 \( 1 - 2.66T + 625T^{2} \)
7 \( 1 - 88.8iT - 2.40e3T^{2} \)
11 \( 1 + 72.9iT - 1.46e4T^{2} \)
13 \( 1 + 173.T + 2.85e4T^{2} \)
17 \( 1 + 423.T + 8.35e4T^{2} \)
19 \( 1 + 153. iT - 1.30e5T^{2} \)
23 \( 1 + 723. iT - 2.79e5T^{2} \)
29 \( 1 - 366.T + 7.07e5T^{2} \)
31 \( 1 - 642. iT - 9.23e5T^{2} \)
37 \( 1 + 535.T + 1.87e6T^{2} \)
41 \( 1 + 2.14e3T + 2.82e6T^{2} \)
43 \( 1 - 615. iT - 3.41e6T^{2} \)
47 \( 1 + 330. iT - 4.87e6T^{2} \)
53 \( 1 - 4.62e3T + 7.89e6T^{2} \)
59 \( 1 - 3.46e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.64e3T + 1.38e7T^{2} \)
67 \( 1 - 6.67e3iT - 2.01e7T^{2} \)
71 \( 1 - 6.44e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.73e3T + 2.83e7T^{2} \)
79 \( 1 - 2.08e3iT - 3.89e7T^{2} \)
83 \( 1 + 9.29e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.70e3T + 6.27e7T^{2} \)
97 \( 1 - 6.97e3T + 8.85e7T^{2} \)
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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

−13.86066177551823994132962936765, −12.71779377391355892325717371068, −11.62730189971921769968517308274, −10.17856392219750353491850988230, −8.943211566468646649600575816031, −8.485311092612526808056998213742, −6.86346671767599495128439089294, −5.82381805014601863785296526215, −4.75580261610172361196528128125, −2.38321031305154646931017364456, 0.14354681859381375727616163243, 1.86256621160982315710344715938, 3.73691590202222030758066679580, 4.74977422036161001394435339382, 7.02463619371487229361843168616, 7.88206519632996341921552737479, 9.483887588484167655949820820104, 10.19407298357460758407028257078, 11.12250149567781120261284885611, 12.20099964591107890578050390895

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