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Properties

Label 2-135-9.4-c1-0-3
Degree $2$
Conductor $135$
Sign $-0.956 + 0.292i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
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  + (−1.04 − 1.80i)2-s + (−1.17 + 2.03i)4-s + (0.5 − 0.866i)5-s + (−2.04 − 3.53i)7-s + 0.734·8-s − 2.08·10-s + (−0.675 − 1.17i)11-s + (−0.324 + 0.561i)13-s + (−4.26 + 7.38i)14-s + (1.58 + 2.74i)16-s + 1.35·17-s + 0.648·19-s + (1.17 + 2.03i)20-s + (−1.41 + 2.44i)22-s + (2.39 − 4.14i)23-s + ⋯
L(s)  = 1  + (−0.737 − 1.27i)2-s + (−0.587 + 1.01i)4-s + (0.223 − 0.387i)5-s + (−0.772 − 1.33i)7-s + 0.259·8-s − 0.659·10-s + (−0.203 − 0.353i)11-s + (−0.0898 + 0.155i)13-s + (−1.13 + 1.97i)14-s + (0.396 + 0.686i)16-s + 0.327·17-s + 0.148·19-s + (0.262 + 0.455i)20-s + (−0.300 + 0.520i)22-s + (0.499 − 0.864i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.956 + 0.292i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ -0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0955859 - 0.638489i\)
\(L(\frac12)\) \(\approx\) \(0.0955859 - 0.638489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.04 + 3.53i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.675 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.324 - 0.561i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 - 0.648T + 19T^{2} \)
23 \( 1 + (-2.39 + 4.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 - 3.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 + (0.0898 - 0.155i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.410 - 0.710i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.45 - 9.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.17T + 53T^{2} \)
59 \( 1 + (-2.08 + 3.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 3.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.07 - 7.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.11T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (5.17 + 8.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.12 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-6.79 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.72168879706398241974917612632, −11.53590345277491164763831567213, −10.55800846576947526319141224132, −9.906111900359676069335126130551, −8.969154011038155017958667532906, −7.69921375364466043029852902976, −6.23644424228524570504671216909, −4.26288910081781160075540917074, −2.89777760502808939558800778458, −0.875053762900632261800193946079, 2.88410088745122794489392423958, 5.35186945034390950931574169800, 6.21594770802415375525296845687, 7.22394298586076588066659082467, 8.396125377184848103919438649662, 9.343240480375625926530965664825, 10.08713165421785826801485754606, 11.74267563200221298400325723821, 12.70876327106755246570692635315, 14.00967171059598550915500280824

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