Properties

Label 2-69-23.3-c1-0-1
Degree $2$
Conductor $69$
Sign $-0.0933 - 0.995i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.319 + 2.22i)2-s + (0.415 − 0.909i)3-s + (−2.91 + 0.855i)4-s + (−1.82 + 2.10i)5-s + (2.15 + 0.632i)6-s + (2.85 − 1.83i)7-s + (−0.966 − 2.11i)8-s + (−0.654 − 0.755i)9-s + (−5.25 − 3.37i)10-s + (0.730 − 5.08i)11-s + (−0.432 + 3.00i)12-s + (−1.83 − 1.17i)13-s + (4.98 + 5.75i)14-s + (1.15 + 2.53i)15-s + (−0.717 + 0.461i)16-s + (2.48 + 0.729i)17-s + ⋯
L(s)  = 1  + (0.225 + 1.57i)2-s + (0.239 − 0.525i)3-s + (−1.45 + 0.427i)4-s + (−0.815 + 0.940i)5-s + (0.879 + 0.258i)6-s + (1.07 − 0.693i)7-s + (−0.341 − 0.747i)8-s + (−0.218 − 0.251i)9-s + (−1.66 − 1.06i)10-s + (0.220 − 1.53i)11-s + (−0.124 + 0.867i)12-s + (−0.508 − 0.327i)13-s + (1.33 + 1.53i)14-s + (0.298 + 0.653i)15-s + (−0.179 + 0.115i)16-s + (0.602 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.0933 - 0.995i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ -0.0933 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666699 + 0.732157i\)
\(L(\frac12)\) \(\approx\) \(0.666699 + 0.732157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (3.18 - 3.58i)T \)
good2 \( 1 + (-0.319 - 2.22i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (1.82 - 2.10i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-2.85 + 1.83i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.730 + 5.08i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (1.83 + 1.17i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-2.48 - 0.729i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (2.21 - 0.651i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (5.12 + 1.50i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-3.71 - 8.14i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (1.43 + 1.65i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (0.199 - 0.230i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-3.81 + 8.35i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 4.99T + 47T^{2} \)
53 \( 1 + (-6.89 + 4.43i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-4.02 - 2.58i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-2.43 - 5.32i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.130 - 0.906i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-0.189 - 1.31i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (9.77 - 2.86i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-11.9 - 7.70i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (1.75 + 2.03i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (0.324 - 0.709i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-2.43 + 2.80i)T + (-13.8 - 96.0i)T^{2} \)
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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

−14.82858695691295216045295762575, −14.31563296010832075814311736864, −13.51148506666838372940572480201, −11.74072080819603617070519647855, −10.71407476117809867381029228571, −8.484694102770456717667977819851, −7.76860915033260545655944799291, −6.95449845366076246164790959619, −5.58224649750323726850935943673, −3.75935971435840361567807914181, 2.11059027743469470837226898081, 4.24049378106564740804285567044, 4.85057124130277007294785231597, 7.86713253398324142587264498559, 9.087197026818617720178628647371, 10.04148090033379861654888926641, 11.45729342889037642579160109746, 12.06138175390902576149554096780, 12.84300676242044580275289664689, 14.45067988934701348448806385516

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