Properties

Label 2-135-5.4-c1-0-1
Degree $2$
Conductor $135$
Sign $-0.316 - 0.948i$
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  + 1.41i·2-s + (−2.12 + 0.707i)5-s + 3i·7-s + 2.82i·8-s + (−1.00 − 3i)10-s + 4.24·11-s − 3i·13-s − 4.24·14-s − 4.00·16-s − 2.82i·17-s − 19-s + 6i·22-s − 7.07i·23-s + (3.99 − 3i)25-s + 4.24·26-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.948 + 0.316i)5-s + 1.13i·7-s + 0.999i·8-s + (−0.316 − 0.948i)10-s + 1.27·11-s − 0.832i·13-s − 1.13·14-s − 1.00·16-s − 0.685i·17-s − 0.229·19-s + 1.27i·22-s − 1.47i·23-s + (0.799 − 0.600i)25-s + 0.832·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627599 + 0.870747i\)
\(L(\frac12)\) \(\approx\) \(0.627599 + 0.870747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.707i)T \)
good2 \( 1 - 1.41iT - 2T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 7.07iT - 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 9iT - 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 9iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 1.41iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 3iT - 97T^{2} \)
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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

−13.92313867393577470793387355725, −12.16983938572200458936797044442, −11.81699157361380268074658552483, −10.56192812178048862507592848038, −8.875409603854670975065563328688, −8.201017349929793109061297698646, −6.96175487110205739025560028789, −6.12665419268401875548806668363, −4.71949257514534425789498584027, −2.84906386063257071396066027475, 1.34607187220342934847295857564, 3.62318633195932765068653292906, 4.27148609216619931026052515335, 6.58304298289204776394057602223, 7.49608998997687987425970719676, 8.960327204148331412512532930119, 10.05893226594612080803856190738, 11.17438047543866167837708759686, 11.70378633207266595458532030280, 12.63902306912569618750312023050

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