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Properties

Label 2-136-136.11-c1-0-2
Degree $2$
Conductor $136$
Sign $-0.168 - 0.985i$
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  + (−1.30 − 0.541i)2-s + (0.675 + 3.39i)3-s + (1.41 + 1.41i)4-s + (0.955 − 4.80i)6-s + (−1.08 − 2.61i)8-s + (−8.30 + 3.44i)9-s + (1.57 + 0.312i)11-s + (−3.84 + 5.75i)12-s + 4i·16-s + (−1.68 − 3.76i)17-s + 12.7·18-s + (6.84 + 2.83i)19-s + (−1.88 − 1.25i)22-s + (8.14 − 5.44i)24-s + (1.91 + 4.61i)25-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.390 + 1.96i)3-s + (0.707 + 0.707i)4-s + (0.390 − 1.96i)6-s + (−0.382 − 0.923i)8-s + (−2.76 + 1.14i)9-s + (0.474 + 0.0943i)11-s + (−1.11 + 1.66i)12-s + i·16-s + (−0.409 − 0.912i)17-s + 2.99·18-s + (1.57 + 0.650i)19-s + (−0.402 − 0.268i)22-s + (1.66 − 1.11i)24-s + (0.382 + 0.923i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.168 - 0.985i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ -0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508778 + 0.603026i\)
\(L(\frac12)\) \(\approx\) \(0.508778 + 0.603026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.541i)T \)
17 \( 1 + (1.68 + 3.76i)T \)
good3 \( 1 + (-0.675 - 3.39i)T + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.57 - 0.312i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 13iT^{2} \)
19 \( 1 + (-6.84 - 2.83i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-6.72 + 4.49i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-5.53 + 2.29i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-4.19 - 10.1i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (23.3 - 56.3i)T^{2} \)
67 \( 1 + 2.48iT - 67T^{2} \)
71 \( 1 + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (13.5 + 9.03i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (3.63 - 8.77i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (10.2 + 10.2i)T + 89iT^{2} \)
97 \( 1 + (0.618 - 0.925i)T + (-37.1 - 89.6i)T^{2} \)
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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

−13.73452884637611622022702177624, −11.89656850720598360653351519068, −11.15429104198187208420053947765, −10.20133035859765173432798937146, −9.399077828099794400760762435537, −8.829166646652959060868655696595, −7.48237494572270788406481120689, −5.50946672377319158111729834831, −4.02124759157911040754029888454, −2.90399824475274456694540536329, 1.15460387315152924028960478046, 2.67180536808398686720395173741, 5.83510517290556170062165298703, 6.72336217301712409735714041931, 7.59445139820643424462881503048, 8.465392507107217964688184482719, 9.374025057718411006874302573697, 11.09604763624973226981931886834, 11.92234939699969219854286403340, 12.91960145134203921515041312241

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