Properties

Label 2-69-23.11-c6-0-8
Degree $2$
Conductor $69$
Sign $0.358 - 0.933i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.62 − 4.25i)2-s + (2.21 − 15.4i)3-s + (−0.862 + 1.88i)4-s + (21.5 + 73.4i)5-s + (−50.9 − 111. i)6-s + (−210. + 182. i)7-s + (73.9 + 514. i)8-s + (−233. − 68.4i)9-s + (455. + 394. i)10-s + (−708. + 1.10e3i)11-s + (27.2 + 17.4i)12-s + (−434. + 501. i)13-s + (−618. + 2.10e3i)14-s + (1.18e3 − 169. i)15-s + (2.59e3 + 2.99e3i)16-s + (−1.88e3 + 861. i)17-s + ⋯
L(s)  = 1  + (0.827 − 0.531i)2-s + (0.0821 − 0.571i)3-s + (−0.0134 + 0.0294i)4-s + (0.172 + 0.587i)5-s + (−0.235 − 0.516i)6-s + (−0.614 + 0.532i)7-s + (0.144 + 1.00i)8-s + (−0.319 − 0.0939i)9-s + (0.455 + 0.394i)10-s + (−0.532 + 0.828i)11-s + (0.0157 + 0.0101i)12-s + (−0.197 + 0.228i)13-s + (−0.225 + 0.767i)14-s + (0.350 − 0.0503i)15-s + (0.632 + 0.730i)16-s + (−0.383 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ 0.358 - 0.933i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.62409 + 1.11584i\)
\(L(\frac12)\) \(\approx\) \(1.62409 + 1.11584i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.21 + 15.4i)T \)
23 \( 1 + (-1.02e4 - 6.61e3i)T \)
good2 \( 1 + (-6.62 + 4.25i)T + (26.5 - 58.2i)T^{2} \)
5 \( 1 + (-21.5 - 73.4i)T + (-1.31e4 + 8.44e3i)T^{2} \)
7 \( 1 + (210. - 182. i)T + (1.67e4 - 1.16e5i)T^{2} \)
11 \( 1 + (708. - 1.10e3i)T + (-7.35e5 - 1.61e6i)T^{2} \)
13 \( 1 + (434. - 501. i)T + (-6.86e5 - 4.77e6i)T^{2} \)
17 \( 1 + (1.88e3 - 861. i)T + (1.58e7 - 1.82e7i)T^{2} \)
19 \( 1 + (-567. - 259. i)T + (3.08e7 + 3.55e7i)T^{2} \)
29 \( 1 + (5.55e3 + 1.21e4i)T + (-3.89e8 + 4.49e8i)T^{2} \)
31 \( 1 + (-1.05e3 - 7.33e3i)T + (-8.51e8 + 2.50e8i)T^{2} \)
37 \( 1 + (5.72e3 - 1.95e4i)T + (-2.15e9 - 1.38e9i)T^{2} \)
41 \( 1 + (4.14e4 - 1.21e4i)T + (3.99e9 - 2.56e9i)T^{2} \)
43 \( 1 + (-3.81e4 - 5.48e3i)T + (6.06e9 + 1.78e9i)T^{2} \)
47 \( 1 + 2.05e4T + 1.07e10T^{2} \)
53 \( 1 + (6.38e4 - 5.53e4i)T + (3.15e9 - 2.19e10i)T^{2} \)
59 \( 1 + (-7.89e4 + 9.11e4i)T + (-6.00e9 - 4.17e10i)T^{2} \)
61 \( 1 + (2.16e5 - 3.11e4i)T + (4.94e10 - 1.45e10i)T^{2} \)
67 \( 1 + (1.15e5 + 1.79e5i)T + (-3.75e10 + 8.22e10i)T^{2} \)
71 \( 1 + (-4.83e5 + 3.11e5i)T + (5.32e10 - 1.16e11i)T^{2} \)
73 \( 1 + (-1.08e5 + 2.37e5i)T + (-9.91e10 - 1.14e11i)T^{2} \)
79 \( 1 + (2.99e5 + 2.59e5i)T + (3.45e10 + 2.40e11i)T^{2} \)
83 \( 1 + (1.36e4 - 4.63e4i)T + (-2.75e11 - 1.76e11i)T^{2} \)
89 \( 1 + (-9.96e5 - 1.43e5i)T + (4.76e11 + 1.40e11i)T^{2} \)
97 \( 1 + (-4.51e5 - 1.53e6i)T + (-7.00e11 + 4.50e11i)T^{2} \)
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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

−13.45037619321334421813108470901, −12.70564335392528921823425864562, −11.86255445648303336083851815435, −10.65105565133586878445106015965, −9.180917666216639912667254518846, −7.70532680143758058832864362816, −6.39879431869849324571102903677, −4.91060778870196971993955735157, −3.18392602476293546142743702844, −2.15890036045488171743092631669, 0.57184373171069688445336946639, 3.31304965524124188947495996234, 4.68615071799672309780792213699, 5.66919409831491113348791594724, 7.01860621193607876429696017780, 8.745402902150919947307181381203, 9.881357358766521871934585600011, 10.93467370554620098941557161225, 12.73140670220572852940153860731, 13.38534071268215598202285445396

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