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Properties

Label 2-138-23.22-c6-0-14
Degree $2$
Conductor $138$
Sign $0.848 + 0.529i$
Analytic cond. $31.7474$
Root an. cond. $5.63448$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65·2-s + 15.5·3-s + 32.0·4-s − 203. i·5-s − 88.1·6-s + 33.7i·7-s − 181.·8-s + 243·9-s + 1.15e3i·10-s + 2.59e3i·11-s + 498.·12-s + 4.00e3·13-s − 190. i·14-s − 3.17e3i·15-s + 1.02e3·16-s + 3.23e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.500·4-s − 1.63i·5-s − 0.408·6-s + 0.0983i·7-s − 0.353·8-s + 0.333·9-s + 1.15i·10-s + 1.94i·11-s + 0.288·12-s + 1.82·13-s − 0.0695i·14-s − 0.941i·15-s + 0.250·16-s + 0.657i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(31.7474\)
Root analytic conductor: \(5.63448\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.970753281\)
\(L(\frac12)\) \(\approx\) \(1.970753281\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65T \)
3 \( 1 - 15.5T \)
23 \( 1 + (-1.03e4 - 6.44e3i)T \)
good5 \( 1 + 203. iT - 1.56e4T^{2} \)
7 \( 1 - 33.7iT - 1.17e5T^{2} \)
11 \( 1 - 2.59e3iT - 1.77e6T^{2} \)
13 \( 1 - 4.00e3T + 4.82e6T^{2} \)
17 \( 1 - 3.23e3iT - 2.41e7T^{2} \)
19 \( 1 + 5.48e3iT - 4.70e7T^{2} \)
29 \( 1 - 3.00e4T + 5.94e8T^{2} \)
31 \( 1 - 2.74e3T + 8.87e8T^{2} \)
37 \( 1 + 8.07e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.21e5T + 4.75e9T^{2} \)
43 \( 1 + 3.18e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.11e5T + 1.07e10T^{2} \)
53 \( 1 + 6.70e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.59e5T + 4.21e10T^{2} \)
61 \( 1 + 1.93e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.56e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.00e5T + 1.28e11T^{2} \)
73 \( 1 - 5.90e5T + 1.51e11T^{2} \)
79 \( 1 - 4.65e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.15e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.51e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.30e6iT - 8.32e11T^{2} \)
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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

−12.16322977792033859800597449445, −10.73670688991539623441069345769, −9.502850178034080246684192841180, −8.861348519243389815925737156190, −8.084244614992626115474020664890, −6.80158938371284284682340377516, −5.16688896076700323736447325410, −3.96544323348632923571552063699, −1.94812575660711175595812350488, −0.973251289986033498535699720176, 1.03256393399489049248104166647, 2.88459444741195826757073110362, 3.48673294121897897521954693308, 6.05540050610358620559559276915, 6.77042757758678616582501796600, 8.128866230704298997117077693713, 8.791506945068506143361661994837, 10.30189489726891199761077108523, 10.86369142320940971789774074221, 11.68348636030787598689497750553

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