Properties

Label 2-138-23.3-c3-0-1
Degree $2$
Conductor $138$
Sign $-0.936 + 0.351i$
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  + (0.284 + 1.97i)2-s + (1.24 − 2.72i)3-s + (−3.83 + 1.12i)4-s + (−7.03 + 8.11i)5-s + (5.75 + 1.69i)6-s + (−5.50 + 3.53i)7-s + (−3.32 − 7.27i)8-s + (−5.89 − 6.80i)9-s + (−18.0 − 11.6i)10-s + (−0.141 + 0.987i)11-s + (−1.70 + 11.8i)12-s + (−47.0 − 30.2i)13-s + (−8.57 − 9.89i)14-s + (13.3 + 29.2i)15-s + (13.4 − 8.65i)16-s + (−63.0 − 18.5i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (−0.628 + 0.725i)5-s + (0.391 + 0.115i)6-s + (−0.297 + 0.191i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.571 − 0.367i)10-s + (−0.00389 + 0.0270i)11-s + (−0.0410 + 0.285i)12-s + (−1.00 − 0.645i)13-s + (−0.163 − 0.188i)14-s + (0.230 + 0.504i)15-s + (0.210 − 0.135i)16-s + (−0.899 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(8.14226\)
Root analytic conductor: \(2.85346\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3/2),\ -0.936 + 0.351i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0508761 - 0.280452i\)
\(L(\frac12)\) \(\approx\) \(0.0508761 - 0.280452i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.284 - 1.97i)T \)
3 \( 1 + (-1.24 + 2.72i)T \)
23 \( 1 + (34.5 - 104. i)T \)
good5 \( 1 + (7.03 - 8.11i)T + (-17.7 - 123. i)T^{2} \)
7 \( 1 + (5.50 - 3.53i)T + (142. - 312. i)T^{2} \)
11 \( 1 + (0.141 - 0.987i)T + (-1.27e3 - 374. i)T^{2} \)
13 \( 1 + (47.0 + 30.2i)T + (912. + 1.99e3i)T^{2} \)
17 \( 1 + (63.0 + 18.5i)T + (4.13e3 + 2.65e3i)T^{2} \)
19 \( 1 + (132. - 38.8i)T + (5.77e3 - 3.70e3i)T^{2} \)
29 \( 1 + (-164. - 48.4i)T + (2.05e4 + 1.31e4i)T^{2} \)
31 \( 1 + (101. + 221. i)T + (-1.95e4 + 2.25e4i)T^{2} \)
37 \( 1 + (-84.3 - 97.3i)T + (-7.20e3 + 5.01e4i)T^{2} \)
41 \( 1 + (-162. + 187. i)T + (-9.80e3 - 6.82e4i)T^{2} \)
43 \( 1 + (16.3 - 35.7i)T + (-5.20e4 - 6.00e4i)T^{2} \)
47 \( 1 - 269.T + 1.03e5T^{2} \)
53 \( 1 + (-19.3 + 12.4i)T + (6.18e4 - 1.35e5i)T^{2} \)
59 \( 1 + (-297. - 191. i)T + (8.53e4 + 1.86e5i)T^{2} \)
61 \( 1 + (-103. - 226. i)T + (-1.48e5 + 1.71e5i)T^{2} \)
67 \( 1 + (-89.7 - 624. i)T + (-2.88e5 + 8.47e4i)T^{2} \)
71 \( 1 + (-45.5 - 317. i)T + (-3.43e5 + 1.00e5i)T^{2} \)
73 \( 1 + (526. - 154. i)T + (3.27e5 - 2.10e5i)T^{2} \)
79 \( 1 + (932. + 599. i)T + (2.04e5 + 4.48e5i)T^{2} \)
83 \( 1 + (228. + 264. i)T + (-8.13e4 + 5.65e5i)T^{2} \)
89 \( 1 + (-226. + 495. i)T + (-4.61e5 - 5.32e5i)T^{2} \)
97 \( 1 + (791. - 913. i)T + (-1.29e5 - 9.03e5i)T^{2} \)
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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

−13.27995514280830969910686353086, −12.49718625803586829259952600764, −11.36866608741362789669076383376, −10.11923329844927598258721554289, −8.826201253850929126794989042200, −7.72343152567718167953329210202, −6.97497726459573515197873061050, −5.84764612812253906733166738499, −4.16682103963497525603755955477, −2.64220311587882412446372886142, 0.12599989512388693110154248658, 2.39993760888366817120611008883, 4.13810553762740187539806551780, 4.74422072845925924688410398866, 6.64575415764736823409512981928, 8.314779183355738273930691130536, 9.023733910799051645173960095996, 10.18145030442711998351692184249, 11.08483262714794424988556442412, 12.25326057448342988009368303170

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