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Properties

Label 2-138-69.68-c5-0-34
Degree $2$
Conductor $138$
Sign $-0.957 - 0.289i$
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 + (−15.1 + 3.51i)3-s − 16·4-s + 0.0328·5-s + (14.0 + 60.7i)6-s − 156. i·7-s + 64i·8-s + (218. − 106. i)9-s − 0.131i·10-s + 514.·11-s + (242. − 56.2i)12-s + 415.·13-s − 626.·14-s + (−0.498 + 0.115i)15-s + 256·16-s − 1.76e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.974 + 0.225i)3-s − 0.5·4-s + 0.000586·5-s + (0.159 + 0.688i)6-s − 1.20i·7-s + 0.353i·8-s + (0.898 − 0.439i)9-s − 0.000414i·10-s + 1.28·11-s + (0.487 − 0.112i)12-s + 0.682·13-s − 0.853·14-s + (−0.000571 + 0.000132i)15-s + 0.250·16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.957 - 0.289i$
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: \((2,\ 138,\ (\ :5/2),\ -0.957 - 0.289i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5902692010\)
\(L(\frac12)\) \(\approx\) \(0.5902692010\)
\(L(\frac{7}{2})\) 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 \( 1 + (15.1 - 3.51i)T \)
23 \( 1 + (-1.26e3 + 2.19e3i)T \)
good5 \( 1 - 0.0328T + 3.12e3T^{2} \)
7 \( 1 + 156. iT - 1.68e4T^{2} \)
11 \( 1 - 514.T + 1.61e5T^{2} \)
13 \( 1 - 415.T + 3.71e5T^{2} \)
17 \( 1 + 1.76e3T + 1.41e6T^{2} \)
19 \( 1 + 327. iT - 2.47e6T^{2} \)
29 \( 1 + 1.44e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.82e3T + 2.86e7T^{2} \)
37 \( 1 - 1.69e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.22e4iT - 1.15e8T^{2} \)
43 \( 1 - 2.13e4iT - 1.47e8T^{2} \)
47 \( 1 - 972. iT - 2.29e8T^{2} \)
53 \( 1 + 1.62e4T + 4.18e8T^{2} \)
59 \( 1 + 7.88e3iT - 7.14e8T^{2} \)
61 \( 1 + 3.39e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.62e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.42e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.67e4T + 2.07e9T^{2} \)
79 \( 1 + 1.96e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.52e4T + 3.93e9T^{2} \)
89 \( 1 + 1.16e5T + 5.58e9T^{2} \)
97 \( 1 + 1.26e5iT - 8.58e9T^{2} \)
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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

−11.29306835542715041278371889303, −11.04946214604388521238167876653, −9.891722263161080043536737102522, −8.884194761729909346084968903899, −7.14942166622641643346504078817, −6.18076508584573317614189829731, −4.51412344698314411049644645476, −3.81622381121614983580642205380, −1.49666018695836397692977509276, −0.24779530485627906560655633160, 1.66275924325257195160686405517, 4.01477223637489386831745373705, 5.42885972796144576470438303916, 6.21475221901489444860154989830, 7.16756134713228506495240879225, 8.687655306421455686526302299163, 9.450954426625002327213370176048, 11.04810717670640832252423249309, 11.79361238552624254335580983729, 12.78500146207267724974213650163

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