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Properties

Label 2-136-136.115-c0-0-0
Degree $2$
Conductor $136$
Sign $-0.615 + 0.788i$
Analytic cond. $0.0678728$
Root an. cond. $0.260524$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (−1 − i)3-s − 4-s + (−1 + i)6-s + i·8-s + i·9-s + (1 − i)11-s + (1 + i)12-s + 16-s − i·17-s + 18-s + 2i·19-s + (−1 − i)22-s + (1 − i)24-s + i·25-s + ⋯
L(s)  = 1  − i·2-s + (−1 − i)3-s − 4-s + (−1 + i)6-s + i·8-s + i·9-s + (1 − i)11-s + (1 + i)12-s + 16-s − i·17-s + 18-s + 2i·19-s + (−1 − i)22-s + (1 − i)24-s + i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(0.0678728\)
Root analytic conductor: \(0.260524\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :0),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4499568977\)
\(L(\frac12)\) \(\approx\) \(0.4499568977\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
17 \( 1 + iT \)
good3 \( 1 + (1 + i)T + iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-1 + i)T - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−12.85715835520574200022236771097, −11.76561960022422668253953730470, −11.58875341989955402464879387225, −10.31820460079360129722653774897, −9.083110844705150311940738088582, −7.80649168776885473376423669888, −6.35973643123559432573863254683, −5.33625654831850659557344740679, −3.55644053736731572668032407830, −1.42085106340463447177544392325, 4.12555615364533978441904967912, 4.89202963436691088873700380710, 6.15309186342952232387332769817, 7.09243364214847237611653603262, 8.718488301401035354803204082389, 9.663511030670705522782666378170, 10.57382924601604411035752812235, 11.77449855381196807410731862227, 12.83260234817688541120987881577, 14.10829379018122111854593772233

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