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Properties

Label 2-69-69.68-c5-0-24
Degree $2$
Conductor $69$
Sign $0.948 + 0.317i$
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  + 9.59i·2-s + (−8.65 + 12.9i)3-s − 60.0·4-s + 42.3·5-s + (−124. − 83.0i)6-s − 238. i·7-s − 269. i·8-s + (−93.1 − 224. i)9-s + 406. i·10-s − 261.·11-s + (519. − 778. i)12-s − 36.2·13-s + 2.29e3·14-s + (−366. + 548. i)15-s + 662.·16-s − 1.88e3·17-s + ⋯
L(s)  = 1  + 1.69i·2-s + (−0.555 + 0.831i)3-s − 1.87·4-s + 0.757·5-s + (−1.41 − 0.941i)6-s − 1.84i·7-s − 1.48i·8-s + (−0.383 − 0.923i)9-s + 1.28i·10-s − 0.651·11-s + (1.04 − 1.56i)12-s − 0.0594·13-s + 3.12·14-s + (−0.420 + 0.629i)15-s + 0.646·16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.948 + 0.317i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 0.948 + 0.317i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.365691 - 0.0595668i\)
\(L(\frac12)\) \(\approx\) \(0.365691 - 0.0595668i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.65 - 12.9i)T \)
23 \( 1 + (-666. - 2.44e3i)T \)
good2 \( 1 - 9.59iT - 32T^{2} \)
5 \( 1 - 42.3T + 3.12e3T^{2} \)
7 \( 1 + 238. iT - 1.68e4T^{2} \)
11 \( 1 + 261.T + 1.61e5T^{2} \)
13 \( 1 + 36.2T + 3.71e5T^{2} \)
17 \( 1 + 1.88e3T + 1.41e6T^{2} \)
19 \( 1 + 1.30e3iT - 2.47e6T^{2} \)
29 \( 1 + 2.23e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.17e3T + 2.86e7T^{2} \)
37 \( 1 - 1.19e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.39e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.15e4iT - 1.47e8T^{2} \)
47 \( 1 - 104. iT - 2.29e8T^{2} \)
53 \( 1 + 3.04e3T + 4.18e8T^{2} \)
59 \( 1 - 4.22e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.96e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.31e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.48e3iT - 1.80e9T^{2} \)
73 \( 1 + 3.49e3T + 2.07e9T^{2} \)
79 \( 1 - 2.16e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.03e4T + 3.93e9T^{2} \)
89 \( 1 + 2.50e4T + 5.58e9T^{2} \)
97 \( 1 + 8.61e4iT - 8.58e9T^{2} \)
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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

−13.69961669080047612186303304949, −13.45004725319626138212481426344, −11.07804235894121378870867624078, −10.07957209261018346652812051916, −8.988675316528627822138161502721, −7.39950473954919807597074262311, −6.47635715461028449628343180484, −5.18785125461017519951500535889, −4.15759917083037697496215191508, −0.16983571518075808356543045718, 1.88970258970903522934286002050, 2.56041434805179888926254255085, 5.04276543785690119175256223112, 6.20014203253624962567525694475, 8.445330168858722075488390027782, 9.463748444419698230700240466944, 10.79379938158577748914818720787, 11.63387330381144580828659954414, 12.70372445499233936896590222591, 13.01799000887494359915908044507

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