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Properties

Label 2-136-17.4-c1-0-2
Degree $2$
Conductor $136$
Sign $0.788 + 0.615i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
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 − i)3-s + (1 − i)5-s + (−1 − i)7-s + i·9-s + (−1 − i)11-s + 2·13-s − 2i·15-s + (1 + 4i)17-s + 2i·19-s − 2·21-s + (−5 − 5i)23-s + 3i·25-s + (4 + 4i)27-s + (−3 + 3i)29-s + (−3 + 3i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (0.447 − 0.447i)5-s + (−0.377 − 0.377i)7-s + 0.333i·9-s + (−0.301 − 0.301i)11-s + 0.554·13-s − 0.516i·15-s + (0.242 + 0.970i)17-s + 0.458i·19-s − 0.436·21-s + (−1.04 − 1.04i)23-s + 0.600i·25-s + (0.769 + 0.769i)27-s + (−0.557 + 0.557i)29-s + (−0.538 + 0.538i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23058 - 0.423505i\)
\(L(\frac12)\) \(\approx\) \(1.23058 - 0.423505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-1 - 4i)T \)
good3 \( 1 + (-1 + i)T - 3iT^{2} \)
5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + (3 - 3i)T - 31iT^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + (7 + 7i)T + 41iT^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 2iT - 59T^{2} \)
61 \( 1 + (-5 - 5i)T + 61iT^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (-9 + 9i)T - 71iT^{2} \)
73 \( 1 + (-9 + 9i)T - 73iT^{2} \)
79 \( 1 + (5 + 5i)T + 79iT^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−13.12933290606616528754137851184, −12.51324531377935054514399650916, −10.94847219752289816238378446520, −10.02309333859831312135853099222, −8.697082895518350614285870132560, −7.957488517331540430191804289074, −6.64008321229335236841764521920, −5.37251201401205682804507580062, −3.60686433888271190065560754120, −1.82181675459807746972455104933, 2.60296196510728259589461946315, 3.89559203824327489931603815983, 5.57372150522651106357303399456, 6.77767091594969739280626834431, 8.188785056306586810555443673500, 9.463644989670771815975227683399, 9.878645229577452488950612661783, 11.22885367011858156334771716748, 12.32138059191894184951458441355, 13.57162418945354658447240790666

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