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Properties

Label 2-69-69.68-c5-0-18
Degree $2$
Conductor $69$
Sign $0.664 + 0.747i$
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  − 8.75i·2-s + (10.6 + 11.3i)3-s − 44.7·4-s + 43.1·5-s + (99.7 − 93.2i)6-s + 151. i·7-s + 111. i·8-s + (−16.4 + 242. i)9-s − 378. i·10-s + 610.·11-s + (−476. − 509. i)12-s + 132.·13-s + 1.32e3·14-s + (459. + 491. i)15-s − 454.·16-s + 1.72e3·17-s + ⋯
L(s)  = 1  − 1.54i·2-s + (0.682 + 0.730i)3-s − 1.39·4-s + 0.772·5-s + (1.13 − 1.05i)6-s + 1.16i·7-s + 0.615i·8-s + (−0.0676 + 0.997i)9-s − 1.19i·10-s + 1.52·11-s + (−0.954 − 1.02i)12-s + 0.217·13-s + 1.80·14-s + (0.527 + 0.564i)15-s − 0.444·16-s + 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.664 + 0.747i$
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: \((2,\ 69,\ (\ :5/2),\ 0.664 + 0.747i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.27716 - 1.02244i\)
\(L(\frac12)\) \(\approx\) \(2.27716 - 1.02244i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-10.6 - 11.3i)T \)
23 \( 1 + (2.53e3 + 62.8i)T \)
good2 \( 1 + 8.75iT - 32T^{2} \)
5 \( 1 - 43.1T + 3.12e3T^{2} \)
7 \( 1 - 151. iT - 1.68e4T^{2} \)
11 \( 1 - 610.T + 1.61e5T^{2} \)
13 \( 1 - 132.T + 3.71e5T^{2} \)
17 \( 1 - 1.72e3T + 1.41e6T^{2} \)
19 \( 1 + 554. iT - 2.47e6T^{2} \)
29 \( 1 + 7.00e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.81e3T + 2.86e7T^{2} \)
37 \( 1 - 1.32e3iT - 6.93e7T^{2} \)
41 \( 1 - 231. iT - 1.15e8T^{2} \)
43 \( 1 - 1.84e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.17e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.89e3T + 4.18e8T^{2} \)
59 \( 1 - 7.71e3iT - 7.14e8T^{2} \)
61 \( 1 - 1.20e3iT - 8.44e8T^{2} \)
67 \( 1 + 4.10e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.19e3iT - 1.80e9T^{2} \)
73 \( 1 + 7.82e4T + 2.07e9T^{2} \)
79 \( 1 - 7.74e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.70e4T + 3.93e9T^{2} \)
89 \( 1 + 1.35e5T + 5.58e9T^{2} \)
97 \( 1 + 7.51e4iT - 8.58e9T^{2} \)
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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

−13.56346224248794545812542145232, −12.19281645819019752255265718154, −11.46002167039048627635376351537, −9.884876305667059677815749671252, −9.590720622877371820869718202139, −8.439546381725947201106276251844, −5.90334875295327508850354707687, −4.18846688651235510400571818578, −2.86722447538850589579704674009, −1.67057256967040280781460067761, 1.31417294505964388513894395385, 3.86572477319502617948569371727, 5.87080779394336247545861870398, 6.81531931009132593957354199625, 7.71509695401491855026257057717, 8.887201748104297349097192329872, 10.00936253651952756135772451500, 12.03445510812582935640071962177, 13.47407265131419031805093941895, 14.23816917977874101832413276683

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