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Properties

Label 2-69-69.68-c5-0-9
Degree $2$
Conductor $69$
Sign $-0.192 - 0.981i$
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  − 1.27i·2-s + (11.5 + 10.5i)3-s + 30.3·4-s − 80.0·5-s + (13.4 − 14.6i)6-s + 196. i·7-s − 79.5i·8-s + (21.7 + 242. i)9-s + 102. i·10-s + 20.8·11-s + (349. + 319. i)12-s − 791.·13-s + 251.·14-s + (−921. − 842. i)15-s + 870.·16-s + 174.·17-s + ⋯
L(s)  = 1  − 0.225i·2-s + (0.738 + 0.674i)3-s + 0.949·4-s − 1.43·5-s + (0.152 − 0.166i)6-s + 1.51i·7-s − 0.439i·8-s + (0.0895 + 0.995i)9-s + 0.323i·10-s + 0.0520·11-s + (0.700 + 0.640i)12-s − 1.29·13-s + 0.342·14-s + (−1.05 − 0.966i)15-s + 0.849·16-s + 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.192 - 0.981i$
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: \((2,\ 69,\ (\ :5/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.16725 + 1.41784i\)
\(L(\frac12)\) \(\approx\) \(1.16725 + 1.41784i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.5 - 10.5i)T \)
23 \( 1 + (-2.03e3 - 1.50e3i)T \)
good2 \( 1 + 1.27iT - 32T^{2} \)
5 \( 1 + 80.0T + 3.12e3T^{2} \)
7 \( 1 - 196. iT - 1.68e4T^{2} \)
11 \( 1 - 20.8T + 1.61e5T^{2} \)
13 \( 1 + 791.T + 3.71e5T^{2} \)
17 \( 1 - 174.T + 1.41e6T^{2} \)
19 \( 1 - 892. iT - 2.47e6T^{2} \)
29 \( 1 - 5.42e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.74e3T + 2.86e7T^{2} \)
37 \( 1 + 1.07e4iT - 6.93e7T^{2} \)
41 \( 1 - 5.10e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.01e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.20e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.84e4T + 4.18e8T^{2} \)
59 \( 1 + 4.96e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.88e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.11e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.17e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.87e4T + 2.07e9T^{2} \)
79 \( 1 - 6.58e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.35e4T + 3.93e9T^{2} \)
89 \( 1 + 1.06e5T + 5.58e9T^{2} \)
97 \( 1 - 5.96e4iT - 8.58e9T^{2} \)
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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

−14.61483111919653191149236421324, −12.49713454609882774610685535511, −11.89373850502386825614638093394, −10.82904160468188482155400253292, −9.447784777740670122531733561280, −8.241060471264952701161477799854, −7.23799488597556586543632042979, −5.22248482218626508766278002393, −3.51275278212094021939767717064, −2.38045720703233040218196021875, 0.74176117776349095136843897689, 2.83863091666490943293159668068, 4.26565089978431770301817704470, 6.86474271182352185478990085552, 7.38199307838912026139701404926, 8.222365605253057292249350971353, 10.12828706657614487514035757947, 11.43091093272932077549842711979, 12.23349137459783710580269890819, 13.51365382753316171628105698792

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