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Properties

Label 2-69-69.68-c5-0-4
Degree $2$
Conductor $69$
Sign $-0.700 + 0.713i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.700 + 0.713i$
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: \((2,\ 69,\ (\ :5/2),\ -0.700 + 0.713i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.246682 - 0.587979i\)
\(L(\frac12)\) \(\approx\) \(0.246682 - 0.587979i\)
\(L(\frac{7}{2})\) 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

−14.83528443708422850970843759284, −13.51264725449046578420832595269, −12.70389981888114933457630465017, −11.27712857303326435590088780381, −9.055537219344741348899582148473, −8.343055362313212581052071949299, −7.44732646441118717819698399376, −6.31689094716702218792124458607, −4.91889660570940879340966932811, −2.68077955187281505368019847039, 0.24949498936858709395961509858, 2.43473978369834269816096661792, 3.75250175348120823900694112781, 4.61724479610303846895160810143, 7.50063753504345958763003634895, 8.700201151852780137200250543004, 10.13495813321060596601848878759, 10.66528096935028504956596740118, 11.60793100173434117885223557036, 13.26180653166741626574972698132

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