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Properties

Label 2-138-69.68-c3-0-8
Degree $2$
Conductor $138$
Sign $0.932 - 0.361i$
Analytic cond. $8.14226$
Root an. cond. $2.85346$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + (4.90 + 1.72i)3-s − 4·4-s − 10.2·5-s + (3.44 − 9.80i)6-s + 24.1i·7-s + 8i·8-s + (21.0 + 16.8i)9-s + 20.5i·10-s + 49.0·11-s + (−19.6 − 6.88i)12-s + 80.1·13-s + 48.3·14-s + (−50.4 − 17.7i)15-s + 16·16-s − 96.2·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.943 + 0.331i)3-s − 0.5·4-s − 0.919·5-s + (0.234 − 0.667i)6-s + 1.30i·7-s + 0.353i·8-s + (0.780 + 0.625i)9-s + 0.650i·10-s + 1.34·11-s + (−0.471 − 0.165i)12-s + 1.71·13-s + 0.923·14-s + (−0.867 − 0.304i)15-s + 0.250·16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.932 - 0.361i$
Analytic conductor: \(8.14226\)
Root analytic conductor: \(2.85346\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3/2),\ 0.932 - 0.361i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.90312 + 0.355695i\)
\(L(\frac12)\) \(\approx\) \(1.90312 + 0.355695i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 + (-4.90 - 1.72i)T \)
23 \( 1 + (-71.6 - 83.8i)T \)
good5 \( 1 + 10.2T + 125T^{2} \)
7 \( 1 - 24.1iT - 343T^{2} \)
11 \( 1 - 49.0T + 1.33e3T^{2} \)
13 \( 1 - 80.1T + 2.19e3T^{2} \)
17 \( 1 + 96.2T + 4.91e3T^{2} \)
19 \( 1 - 82.7iT - 6.85e3T^{2} \)
29 \( 1 + 174. iT - 2.43e4T^{2} \)
31 \( 1 + 119.T + 2.97e4T^{2} \)
37 \( 1 + 361. iT - 5.06e4T^{2} \)
41 \( 1 - 57.8iT - 6.89e4T^{2} \)
43 \( 1 - 314. iT - 7.95e4T^{2} \)
47 \( 1 - 364. iT - 1.03e5T^{2} \)
53 \( 1 - 284.T + 1.48e5T^{2} \)
59 \( 1 - 130. iT - 2.05e5T^{2} \)
61 \( 1 + 613. iT - 2.26e5T^{2} \)
67 \( 1 + 922. iT - 3.00e5T^{2} \)
71 \( 1 + 209. iT - 3.57e5T^{2} \)
73 \( 1 + 693.T + 3.89e5T^{2} \)
79 \( 1 + 881. iT - 4.93e5T^{2} \)
83 \( 1 - 437.T + 5.71e5T^{2} \)
89 \( 1 - 750.T + 7.04e5T^{2} \)
97 \( 1 - 438. iT - 9.12e5T^{2} \)
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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

−12.70838531064569768841201446986, −11.63184949965755816650029482711, −10.98390081231031085491334679196, −9.291550447695660582271594267295, −8.894578010910448416559011404547, −7.88649241692899160676547443131, −6.10149236071089586621102307566, −4.23088994341281426853934666062, −3.44338163620241624078518311927, −1.81097022305210403920735140748, 1.00827490709293146159407269752, 3.65768689000498708225788114795, 4.27290530368105099822357527635, 6.73054041706345417060252024696, 7.06775120842361821897065596672, 8.514186756133075866371326354571, 8.941929112697360436399081483674, 10.59002713306559481800607751868, 11.65112609422302915272444884992, 13.20825588990507173880244018057

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