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Properties

Label 2-68-17.9-c5-0-6
Degree $2$
Conductor $68$
Sign $-0.560 - 0.828i$
Analytic cond. $10.9060$
Root an. cond. $3.30243$
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  + (−26.9 − 11.1i)3-s + (28.5 − 68.9i)5-s + (−85.1 − 205. i)7-s + (431. + 431. i)9-s + (−317. + 131. i)11-s − 702. i·13-s + (−1.54e3 + 1.54e3i)15-s + (−280. + 1.15e3i)17-s + (−687. + 687. i)19-s + 6.49e3i·21-s + (1.26e3 − 525. i)23-s + (−1.72e3 − 1.72e3i)25-s + (−4.09e3 − 9.89e3i)27-s + (339. − 818. i)29-s + (7.45e3 + 3.08e3i)31-s + ⋯
L(s)  = 1  + (−1.73 − 0.716i)3-s + (0.510 − 1.23i)5-s + (−0.656 − 1.58i)7-s + (1.77 + 1.77i)9-s + (−0.790 + 0.327i)11-s − 1.15i·13-s + (−1.76 + 1.76i)15-s + (−0.235 + 0.971i)17-s + (−0.436 + 0.436i)19-s + 3.21i·21-s + (0.500 − 0.207i)23-s + (−0.553 − 0.553i)25-s + (−1.08 − 2.61i)27-s + (0.0748 − 0.180i)29-s + (1.39 + 0.577i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $-0.560 - 0.828i$
Analytic conductor: \(10.9060\)
Root analytic conductor: \(3.30243\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :5/2),\ -0.560 - 0.828i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.179375 + 0.337866i\)
\(L(\frac12)\) \(\approx\) \(0.179375 + 0.337866i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (280. - 1.15e3i)T \)
good3 \( 1 + (26.9 + 11.1i)T + (171. + 171. i)T^{2} \)
5 \( 1 + (-28.5 + 68.9i)T + (-2.20e3 - 2.20e3i)T^{2} \)
7 \( 1 + (85.1 + 205. i)T + (-1.18e4 + 1.18e4i)T^{2} \)
11 \( 1 + (317. - 131. i)T + (1.13e5 - 1.13e5i)T^{2} \)
13 \( 1 + 702. iT - 3.71e5T^{2} \)
19 \( 1 + (687. - 687. i)T - 2.47e6iT^{2} \)
23 \( 1 + (-1.26e3 + 525. i)T + (4.55e6 - 4.55e6i)T^{2} \)
29 \( 1 + (-339. + 818. i)T + (-1.45e7 - 1.45e7i)T^{2} \)
31 \( 1 + (-7.45e3 - 3.08e3i)T + (2.02e7 + 2.02e7i)T^{2} \)
37 \( 1 + (-3.79e3 - 1.57e3i)T + (4.90e7 + 4.90e7i)T^{2} \)
41 \( 1 + (3.94e3 + 9.53e3i)T + (-8.19e7 + 8.19e7i)T^{2} \)
43 \( 1 + (8.41e3 + 8.41e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 189. iT - 2.29e8T^{2} \)
53 \( 1 + (4.02e3 - 4.02e3i)T - 4.18e8iT^{2} \)
59 \( 1 + (-1.40e4 - 1.40e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (4.79e3 + 1.15e4i)T + (-5.97e8 + 5.97e8i)T^{2} \)
67 \( 1 + 4.22e4T + 1.35e9T^{2} \)
71 \( 1 + (2.92e3 + 1.21e3i)T + (1.27e9 + 1.27e9i)T^{2} \)
73 \( 1 + (3.35e3 - 8.09e3i)T + (-1.46e9 - 1.46e9i)T^{2} \)
79 \( 1 + (6.26e4 - 2.59e4i)T + (2.17e9 - 2.17e9i)T^{2} \)
83 \( 1 + (-4.60e4 + 4.60e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 7.44e4iT - 5.58e9T^{2} \)
97 \( 1 + (-2.95e4 + 7.12e4i)T + (-6.07e9 - 6.07e9i)T^{2} \)
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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

−13.03364771665193700780931272623, −12.25031577551928468339742436617, −10.57787909519902933490721940722, −10.22216640283561647779126550086, −8.019059475137870992891676199010, −6.77552312531123074124457657112, −5.62563399758774428965876376110, −4.53270585631958778513339762637, −1.23455001013674282505430328239, −0.23874748426547069587349176150, 2.75141662609860699566655987087, 4.90929689817342915289539486695, 6.08971610747893399089336337329, 6.69302927887234334855984090197, 9.268840912956899315619712696300, 10.13546951228062967730328903353, 11.27490703586439879817533866842, 11.84277524758278410004886254414, 13.20483740311754105453064696444, 14.88405029831181546702726301192

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