Properties

Label 2-136-17.8-c1-0-1
Degree $2$
Conductor $136$
Sign $0.815 - 0.578i$
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  + (−0.292 + 0.707i)3-s + (1.70 + 0.707i)5-s + (−0.292 + 0.121i)7-s + (1.70 + 1.70i)9-s + (0.292 + 0.707i)11-s + (−1 + 0.999i)15-s + (1 − 4i)17-s + (−1.58 + 1.58i)19-s − 0.242i·21-s + (−2.53 − 6.12i)23-s + (−1.12 − 1.12i)25-s + (−3.82 + 1.58i)27-s + (−5.12 − 2.12i)29-s + (0.878 − 2.12i)31-s − 0.585·33-s + ⋯
L(s)  = 1  + (−0.169 + 0.408i)3-s + (0.763 + 0.316i)5-s + (−0.110 + 0.0458i)7-s + (0.569 + 0.569i)9-s + (0.0883 + 0.213i)11-s + (−0.258 + 0.258i)15-s + (0.242 − 0.970i)17-s + (−0.363 + 0.363i)19-s − 0.0529i·21-s + (−0.528 − 1.27i)23-s + (−0.224 − 0.224i)25-s + (−0.736 + 0.305i)27-s + (−0.951 − 0.393i)29-s + (0.157 − 0.381i)31-s − 0.101·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10108 + 0.350510i\)
\(L(\frac12)\) \(\approx\) \(1.10108 + 0.350510i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 - 0.707i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.70 - 0.707i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.292 - 0.121i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.292 - 0.707i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 13T^{2} \)
19 \( 1 + (1.58 - 1.58i)T - 19iT^{2} \)
23 \( 1 + (2.53 + 6.12i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (5.12 + 2.12i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.878 + 2.12i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.29 - 0.949i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-7.24 - 7.24i)T + 43iT^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + (-2.17 + 2.17i)T - 53iT^{2} \)
59 \( 1 + (-5.58 - 5.58i)T + 59iT^{2} \)
61 \( 1 + (9.12 - 3.77i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 9.65T + 67T^{2} \)
71 \( 1 + (-3.70 + 8.94i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.36 - 3.05i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-5.46 - 13.1i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (7.24 - 7.24i)T - 83iT^{2} \)
89 \( 1 + 1.65iT - 89T^{2} \)
97 \( 1 + (14.7 + 6.12i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.39958871091039145432193388428, −12.35723454683904325862665090905, −11.10579186374486947590134654985, −10.13257147492482251612701421890, −9.500405471843425007995337275833, −7.991922040003695595499183319479, −6.71979159039829428066635544096, −5.51855675954712629303288004227, −4.22183942878266038100927665634, −2.29727338820350504257485595163, 1.65159012694999088176465622880, 3.78770171404982152271895317798, 5.50894171060124303790079343854, 6.47303129405001764875303400762, 7.69990800604350471163860286566, 9.094579766826229471778090474118, 9.903092684985055607655007149584, 11.11588859776937810530170932988, 12.34016687267705284685288447425, 13.04468226344962214551961478758

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