Properties

Label 2-138-23.22-c6-0-17
Degree $2$
Conductor $138$
Sign $-0.926 + 0.375i$
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 − 33.5i·5-s + 88.1·6-s − 442. i·7-s − 181.·8-s + 243·9-s + 189. i·10-s − 934. i·11-s − 498.·12-s + 905.·13-s + 2.50e3i·14-s + 522. i·15-s + 1.02e3·16-s − 3.77e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.500·4-s − 0.268i·5-s + 0.408·6-s − 1.28i·7-s − 0.353·8-s + 0.333·9-s + 0.189i·10-s − 0.702i·11-s − 0.288·12-s + 0.411·13-s + 0.911i·14-s + 0.154i·15-s + 0.250·16-s − 0.769i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.926 + 0.375i$
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: \((2,\ 138,\ (\ :3),\ -0.926 + 0.375i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7043041798\)
\(L(\frac12)\) \(\approx\) \(0.7043041798\)
\(L(4)\) 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 \( 1 + (-1.12e4 + 4.56e3i)T \)
good5 \( 1 + 33.5iT - 1.56e4T^{2} \)
7 \( 1 + 442. iT - 1.17e5T^{2} \)
11 \( 1 + 934. iT - 1.77e6T^{2} \)
13 \( 1 - 905.T + 4.82e6T^{2} \)
17 \( 1 + 3.77e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.05e3iT - 4.70e7T^{2} \)
29 \( 1 - 1.79e3T + 5.94e8T^{2} \)
31 \( 1 + 3.34e3T + 8.87e8T^{2} \)
37 \( 1 + 4.06e4iT - 2.56e9T^{2} \)
41 \( 1 + 4.45e4T + 4.75e9T^{2} \)
43 \( 1 - 1.04e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.85e5T + 1.07e10T^{2} \)
53 \( 1 + 5.80e4iT - 2.21e10T^{2} \)
59 \( 1 + 3.62e5T + 4.21e10T^{2} \)
61 \( 1 + 3.80e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.31e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.18e5T + 1.28e11T^{2} \)
73 \( 1 + 6.02e3T + 1.51e11T^{2} \)
79 \( 1 - 8.99e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.64e5iT - 3.26e11T^{2} \)
89 \( 1 + 7.70e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.70e6iT - 8.32e11T^{2} \)
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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

−11.23666704169179907249631408442, −10.68372323216129560575513572911, −9.632385628173004327398584657365, −8.418047167417198875573847264788, −7.30993111221016738486213635531, −6.33763747125810134483228847123, −4.89217557918742248621452146696, −3.40242086755529946641906674694, −1.28870744694721526100881695533, −0.33040763626242104765040971613, 1.49685263936140117010532792699, 2.91206568023037498742602292900, 4.90385987730949869566045528516, 6.11919130125663129727097520303, 7.08518552146609088904858134790, 8.481164479036899507356658097563, 9.338324442191681210108294405699, 10.48479509507272648931931817895, 11.39511097560539839118263152464, 12.25498039470709285027225416544

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