Properties

Label 2-69-69.68-c5-0-6
Degree $2$
Conductor $69$
Sign $-0.678 + 0.734i$
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  + 5.41i·2-s + (3.03 + 15.2i)3-s + 2.64·4-s − 41.1·5-s + (−82.8 + 16.4i)6-s − 67.8i·7-s + 187. i·8-s + (−224. + 92.8i)9-s − 223. i·10-s − 718.·11-s + (8.01 + 40.3i)12-s − 17.0·13-s + 367.·14-s + (−125. − 629. i)15-s − 932.·16-s + 1.85e3·17-s + ⋯
L(s)  = 1  + 0.957i·2-s + (0.194 + 0.980i)3-s + 0.0825·4-s − 0.736·5-s + (−0.939 + 0.186i)6-s − 0.523i·7-s + 1.03i·8-s + (−0.924 + 0.381i)9-s − 0.705i·10-s − 1.79·11-s + (0.0160 + 0.0809i)12-s − 0.0279·13-s + 0.501·14-s + (−0.143 − 0.722i)15-s − 0.910·16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.678 + 0.734i$
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: \((2,\ 69,\ (\ :5/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.382206 - 0.873004i\)
\(L(\frac12)\) \(\approx\) \(0.382206 - 0.873004i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.03 - 15.2i)T \)
23 \( 1 + (1.49e3 + 2.05e3i)T \)
good2 \( 1 - 5.41iT - 32T^{2} \)
5 \( 1 + 41.1T + 3.12e3T^{2} \)
7 \( 1 + 67.8iT - 1.68e4T^{2} \)
11 \( 1 + 718.T + 1.61e5T^{2} \)
13 \( 1 + 17.0T + 3.71e5T^{2} \)
17 \( 1 - 1.85e3T + 1.41e6T^{2} \)
19 \( 1 - 1.13e3iT - 2.47e6T^{2} \)
29 \( 1 - 1.74e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.91e3T + 2.86e7T^{2} \)
37 \( 1 - 9.22e3iT - 6.93e7T^{2} \)
41 \( 1 - 5.01e3iT - 1.15e8T^{2} \)
43 \( 1 - 5.50e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.05e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.66e4T + 4.18e8T^{2} \)
59 \( 1 - 3.00e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.03e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.26e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.28e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.04e4T + 2.07e9T^{2} \)
79 \( 1 - 3.71e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.57e4T + 3.93e9T^{2} \)
89 \( 1 + 1.07e5T + 5.58e9T^{2} \)
97 \( 1 + 6.92e4iT - 8.58e9T^{2} \)
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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

−14.76702579425694410932134810411, −13.83784116990488647650182673000, −12.14152907662444035458618383310, −10.85180941122530642125114245380, −10.02777714107427381858969343771, −8.088003733395986842587135356958, −7.78405655536692300019856726491, −5.87948570469647000995107139633, −4.68665641069440468131000132951, −3.01093419074907673163536614040, 0.38960941713970809311380473620, 2.19133378192585006996658685223, 3.32649546369279729745994590300, 5.61749678857529279245899218652, 7.33002143813543364172679519988, 8.138711793752213970843572406024, 9.842630448362559882550877399362, 11.12536476028562414312455940320, 12.03527118658473244037211974460, 12.69620028210346589009463769053

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