Properties

Label 2-136-17.3-c2-0-6
Degree $2$
Conductor $136$
Sign $0.999 - 0.0378i$
Analytic cond. $3.70573$
Root an. cond. $1.92502$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.38 + 0.872i)3-s + (−0.381 − 0.571i)5-s + (4.41 − 6.61i)7-s + (10.1 + 4.21i)9-s + (1.23 + 6.22i)11-s + (−2.83 − 2.83i)13-s + (−1.17 − 2.84i)15-s + (4.11 + 16.4i)17-s + (−27.5 + 11.3i)19-s + (25.1 − 25.1i)21-s + (−8.11 + 1.61i)23-s + (9.38 − 22.6i)25-s + (7.51 + 5.02i)27-s + (32.0 − 21.3i)29-s + (−5.66 + 28.4i)31-s + ⋯
L(s)  = 1  + (1.46 + 0.290i)3-s + (−0.0763 − 0.114i)5-s + (0.631 − 0.944i)7-s + (1.13 + 0.468i)9-s + (0.112 + 0.566i)11-s + (−0.218 − 0.218i)13-s + (−0.0784 − 0.189i)15-s + (0.241 + 0.970i)17-s + (−1.44 + 0.599i)19-s + (1.19 − 1.19i)21-s + (−0.352 + 0.0701i)23-s + (0.375 − 0.906i)25-s + (0.278 + 0.185i)27-s + (1.10 − 0.737i)29-s + (−0.182 + 0.918i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0378i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.999 - 0.0378i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ 0.999 - 0.0378i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.20202 + 0.0416839i\)
\(L(\frac12)\) \(\approx\) \(2.20202 + 0.0416839i\)
\(L(2)\) 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 \( 1 + (-4.11 - 16.4i)T \)
good3 \( 1 + (-4.38 - 0.872i)T + (8.31 + 3.44i)T^{2} \)
5 \( 1 + (0.381 + 0.571i)T + (-9.56 + 23.0i)T^{2} \)
7 \( 1 + (-4.41 + 6.61i)T + (-18.7 - 45.2i)T^{2} \)
11 \( 1 + (-1.23 - 6.22i)T + (-111. + 46.3i)T^{2} \)
13 \( 1 + (2.83 + 2.83i)T + 169iT^{2} \)
19 \( 1 + (27.5 - 11.3i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (8.11 - 1.61i)T + (488. - 202. i)T^{2} \)
29 \( 1 + (-32.0 + 21.3i)T + (321. - 776. i)T^{2} \)
31 \( 1 + (5.66 - 28.4i)T + (-887. - 367. i)T^{2} \)
37 \( 1 + (21.7 + 4.33i)T + (1.26e3 + 523. i)T^{2} \)
41 \( 1 + (14.7 - 22.0i)T + (-643. - 1.55e3i)T^{2} \)
43 \( 1 + (59.4 + 24.6i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (49.0 + 49.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (10.3 - 4.28i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-2.40 + 5.80i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (21.7 + 14.5i)T + (1.42e3 + 3.43e3i)T^{2} \)
67 \( 1 - 47.9iT - 4.48e3T^{2} \)
71 \( 1 + (-131. - 26.1i)T + (4.65e3 + 1.92e3i)T^{2} \)
73 \( 1 + (-58.4 - 87.4i)T + (-2.03e3 + 4.92e3i)T^{2} \)
79 \( 1 + (22.7 + 114. i)T + (-5.76e3 + 2.38e3i)T^{2} \)
83 \( 1 + (-51.5 - 124. i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-97.2 + 97.2i)T - 7.92e3iT^{2} \)
97 \( 1 + (-140. + 93.8i)T + (3.60e3 - 8.69e3i)T^{2} \)
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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

−13.20453152375007100896511545922, −12.16618045780656895154373001105, −10.53251226380680530375067111582, −9.968164556630564328489568419072, −8.441961246752816637749667172454, −8.097734817745682695471798169104, −6.70805545381966935417815120111, −4.61653713039220472428409654476, −3.65917909336286090567493747959, −1.95064106871897815658469386548, 2.07270211952659904123061078696, 3.22537631856840325949497069376, 4.93948911336888000772644383886, 6.64542161808518483617147016894, 7.951336730351271153689819789307, 8.683999807281033895654539140774, 9.450808176732556136587747402099, 10.99820059726668546211512388418, 12.07410704627123281259113664391, 13.17779898114406782736719603804

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