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Properties

Label 2-138-69.68-c5-0-38
Degree $2$
Conductor $138$
Sign $-0.745 - 0.666i$
Analytic cond. $22.1329$
Root an. cond. $4.70456$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + (2.23 − 15.4i)3-s − 16·4-s + 58.1·5-s + (−61.7 − 8.94i)6-s − 20.4i·7-s + 64i·8-s + (−233. − 68.9i)9-s − 232. i·10-s − 145.·11-s + (−35.7 + 246. i)12-s − 532.·13-s − 81.8·14-s + (129. − 896. i)15-s + 256·16-s − 836.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.143 − 0.989i)3-s − 0.5·4-s + 1.03·5-s + (−0.699 − 0.101i)6-s − 0.157i·7-s + 0.353i·8-s + (−0.958 − 0.283i)9-s − 0.735i·10-s − 0.363·11-s + (−0.0717 + 0.494i)12-s − 0.874·13-s − 0.111·14-s + (0.149 − 1.02i)15-s + 0.250·16-s − 0.701·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(22.1329\)
Root analytic conductor: \(4.70456\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :5/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.080032845\)
\(L(\frac12)\) \(\approx\) \(1.080032845\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 + (-2.23 + 15.4i)T \)
23 \( 1 + (2.11e3 + 1.40e3i)T \)
good5 \( 1 - 58.1T + 3.12e3T^{2} \)
7 \( 1 + 20.4iT - 1.68e4T^{2} \)
11 \( 1 + 145.T + 1.61e5T^{2} \)
13 \( 1 + 532.T + 3.71e5T^{2} \)
17 \( 1 + 836.T + 1.41e6T^{2} \)
19 \( 1 + 2.40e3iT - 2.47e6T^{2} \)
29 \( 1 - 1.14e3iT - 2.05e7T^{2} \)
31 \( 1 - 162.T + 2.86e7T^{2} \)
37 \( 1 - 9.82e3iT - 6.93e7T^{2} \)
41 \( 1 + 947. iT - 1.15e8T^{2} \)
43 \( 1 + 7.71e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.23e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.82e3T + 4.18e8T^{2} \)
59 \( 1 - 2.32e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.04e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.07e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.89e4T + 2.07e9T^{2} \)
79 \( 1 + 2.63e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.13e4T + 3.93e9T^{2} \)
89 \( 1 + 3.10e3T + 5.58e9T^{2} \)
97 \( 1 + 8.48e4iT - 8.58e9T^{2} \)
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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

−11.77619141541616663766160140393, −10.66717841284119627407353958300, −9.586581734830431096464811972978, −8.624603845665254563304471807669, −7.28496866875133998320261933166, −6.13856210380019381398370764354, −4.82242976013231924658771590438, −2.76933829659682487928107516823, −1.91282273974843532857210651855, −0.33678057734951394798448129715, 2.27090190091884139645250293765, 4.00436827775437880028406980140, 5.33163013846260293633254375145, 6.04114220414159460234715326620, 7.66428392495874875547612662686, 8.841579157702538181165396806172, 9.801084905242805812912730494296, 10.36465172098243651179615614625, 11.88206821724874506721730000290, 13.20780675425125534720945702600

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