Properties

Label 2-69-23.12-c5-0-13
Degree $2$
Conductor $69$
Sign $0.907 + 0.419i$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
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  + (3.85 − 2.47i)2-s + (1.28 − 8.90i)3-s + (−4.57 + 10.0i)4-s + (64.5 − 18.9i)5-s + (−17.1 − 37.4i)6-s + (145. + 167. i)7-s + (28.0 + 195. i)8-s + (−77.7 − 22.8i)9-s + (201. − 232. i)10-s + (−279. − 179. i)11-s + (83.4 + 53.6i)12-s + (645. − 745. i)13-s + (973. + 285. i)14-s + (−86.1 − 598. i)15-s + (360. + 415. i)16-s + (513. + 1.12e3i)17-s + ⋯
L(s)  = 1  + (0.681 − 0.437i)2-s + (0.0821 − 0.571i)3-s + (−0.143 + 0.313i)4-s + (1.15 − 0.338i)5-s + (−0.194 − 0.425i)6-s + (1.11 + 1.29i)7-s + (0.154 + 1.07i)8-s + (−0.319 − 0.0939i)9-s + (0.637 − 0.736i)10-s + (−0.696 − 0.447i)11-s + (0.167 + 0.107i)12-s + (1.05 − 1.22i)13-s + (1.32 + 0.389i)14-s + (−0.0988 − 0.687i)15-s + (0.351 + 0.405i)16-s + (0.430 + 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 0.907 + 0.419i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.06033 - 0.672767i\)
\(L(\frac12)\) \(\approx\) \(3.06033 - 0.672767i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.28 + 8.90i)T \)
23 \( 1 + (1.97e3 + 1.59e3i)T \)
good2 \( 1 + (-3.85 + 2.47i)T + (13.2 - 29.1i)T^{2} \)
5 \( 1 + (-64.5 + 18.9i)T + (2.62e3 - 1.68e3i)T^{2} \)
7 \( 1 + (-145. - 167. i)T + (-2.39e3 + 1.66e4i)T^{2} \)
11 \( 1 + (279. + 179. i)T + (6.69e4 + 1.46e5i)T^{2} \)
13 \( 1 + (-645. + 745. i)T + (-5.28e4 - 3.67e5i)T^{2} \)
17 \( 1 + (-513. - 1.12e3i)T + (-9.29e5 + 1.07e6i)T^{2} \)
19 \( 1 + (-686. + 1.50e3i)T + (-1.62e6 - 1.87e6i)T^{2} \)
29 \( 1 + (-981. - 2.14e3i)T + (-1.34e7 + 1.55e7i)T^{2} \)
31 \( 1 + (-581. - 4.04e3i)T + (-2.74e7 + 8.06e6i)T^{2} \)
37 \( 1 + (-4.07e3 - 1.19e3i)T + (5.83e7 + 3.74e7i)T^{2} \)
41 \( 1 + (1.73e4 - 5.10e3i)T + (9.74e7 - 6.26e7i)T^{2} \)
43 \( 1 + (-2.69e3 + 1.87e4i)T + (-1.41e8 - 4.14e7i)T^{2} \)
47 \( 1 + 6.07e3T + 2.29e8T^{2} \)
53 \( 1 + (8.26e3 + 9.53e3i)T + (-5.95e7 + 4.13e8i)T^{2} \)
59 \( 1 + (7.07e3 - 8.16e3i)T + (-1.01e8 - 7.07e8i)T^{2} \)
61 \( 1 + (-2.55e3 - 1.77e4i)T + (-8.10e8 + 2.37e8i)T^{2} \)
67 \( 1 + (8.06e3 - 5.18e3i)T + (5.60e8 - 1.22e9i)T^{2} \)
71 \( 1 + (-2.33e4 + 1.50e4i)T + (7.49e8 - 1.64e9i)T^{2} \)
73 \( 1 + (-3.51e3 + 7.69e3i)T + (-1.35e9 - 1.56e9i)T^{2} \)
79 \( 1 + (4.59e4 - 5.29e4i)T + (-4.37e8 - 3.04e9i)T^{2} \)
83 \( 1 + (7.07e4 + 2.07e4i)T + (3.31e9 + 2.12e9i)T^{2} \)
89 \( 1 + (-1.17e4 + 8.18e4i)T + (-5.35e9 - 1.57e9i)T^{2} \)
97 \( 1 + (-1.20e5 + 3.52e4i)T + (7.22e9 - 4.64e9i)T^{2} \)
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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

−13.43252281401008597227930805503, −12.76925214058583159552556862187, −11.76765333825703193907755896341, −10.58917225702073842142304670558, −8.650496229664837258528435073344, −8.203374454141411051944992358843, −5.81831507904670088608516810369, −5.18417736271388683425290643373, −2.93308123656578759695090752619, −1.70576653816831614598301509269, 1.53266493478946403128191285250, 3.99450415904048179744732393362, 5.08606017054551541426560689196, 6.28655847281329958118638372368, 7.71673042848315694251290103137, 9.630266912555440658336814431051, 10.24497876285134486923895108864, 11.42577509529072800699702626454, 13.46401370762189610227456913283, 13.95896586473884766395962806306

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