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Properties

Label 1-69-69.11-r0-0-0
Degree $1$
Conductor $69$
Sign $0.938 + 0.343i$
Analytic cond. $0.320434$
Root an. cond. $0.320434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.142 + 0.989i)10-s + (0.415 + 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (0.654 + 0.755i)19-s + (−0.959 − 0.281i)20-s − 22-s + (0.415 − 0.909i)25-s + (0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.142 + 0.989i)10-s + (0.415 + 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (0.654 + 0.755i)19-s + (−0.959 − 0.281i)20-s − 22-s + (0.415 − 0.909i)25-s + (0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.938 + 0.343i$
Analytic conductor: \(0.320434\)
Root analytic conductor: \(0.320434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (0:\ ),\ 0.938 + 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8137070293 + 0.1443500053i\)
\(L(\frac12)\) \(\approx\) \(0.8137070293 + 0.1443500053i\)
\(L(1)\) \(\approx\) \(0.8899987882 + 0.1783700740i\)
\(L(1)\) \(\approx\) \(0.8899987882 + 0.1783700740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.415 + 0.909i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (-0.841 - 0.540i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.4386147791410499595128836205, −30.64207638158385601014502477587, −29.30966221402373428902859383751, −28.86474140896446252732141626495, −27.515661492908150027673867364197, −26.49108486911712530258013159322, −25.48590216067009675586482521466, −24.2392821159529998611334747988, −22.237290929986178739752452267194, −21.893515339033460014200975964208, −20.80559740804322202611998315545, −19.31668730813322527394509184423, −18.44884710597772030997069226663, −17.56568009291094032797394806449, −16.179385702720742440464692085956, −14.344664753290032013524919808325, −13.411202457013816508026975816819, −11.86866868758031517432604026402, −11.00896056530073663784984794030, −9.48303174191687701011816647827, −8.77079079222485166252144346309, −6.8116732698900913621768697865, −5.12900997259858663300045638222, −3.14446597763067472307726583575, −1.91991516442511312934350038573, 1.44832072438346395236575550296, 4.31714857277121455719485055966, 5.62556278685487834258301686463, 6.97993329426242430598678003245, 8.241228521904833116773598825808, 9.65231083170375161879861039626, 10.48419708069232508111908221468, 12.73031864122734976037167146390, 13.8055479701953317971633802768, 14.88484396162371630586203736460, 16.29536244000798956853389132624, 17.38923505056402476351618676841, 17.85472301328270956332292553345, 19.67336729869393090830112664476, 20.57441416690903346094363877173, 22.26466565918762262961032798013, 23.28765842074798383139453645470, 24.499683093663891267180960927984, 25.27555787193350820013161722797, 26.32920449881814882683333542651, 27.43292586640502810916094668704, 28.45634066315016718725780031915, 29.57837374853558648901023464798, 30.94384749640697463037889074420, 32.5440326308523802384565577873

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