Properties

Label 2-69-69.68-c1-0-3
Degree $2$
Conductor $69$
Sign $0.991 - 0.131i$
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  + 0.935i·2-s + (−0.227 − 1.71i)3-s + 1.12·4-s + (1.60 − 0.212i)6-s + 2.92i·8-s + (−2.89 + 0.781i)9-s + (−0.255 − 1.92i)12-s − 4.88·13-s − 0.488·16-s + (−0.731 − 2.71i)18-s − 4.79i·23-s + (5.02 − 0.665i)24-s − 5·25-s − 4.57i·26-s + (1.99 + 4.79i)27-s + ⋯
L(s)  = 1  + 0.661i·2-s + (−0.131 − 0.991i)3-s + 0.561·4-s + (0.656 − 0.0869i)6-s + 1.03i·8-s + (−0.965 + 0.260i)9-s + (−0.0738 − 0.557i)12-s − 1.35·13-s − 0.122·16-s + (−0.172 − 0.638i)18-s − 0.999i·23-s + (1.02 − 0.135i)24-s − 25-s − 0.896i·26-s + (0.384 + 0.922i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.991 - 0.131i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ 0.991 - 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.956967 + 0.0631074i\)
\(L(\frac12)\) \(\approx\) \(0.956967 + 0.0631074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.227 + 1.71i)T \)
23 \( 1 + 4.79iT \)
good2 \( 1 - 0.935iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
29 \( 1 - 8.43iT - 29T^{2} \)
31 \( 1 - 11.1T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 12.1iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 6.55iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 9.59iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 + 7.61T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−14.70986538228289658088489890541, −13.90525843992720231666043891769, −12.45215398479418620567206413169, −11.75426659891393102706861973863, −10.40307041637382338142299505049, −8.556597545613138012436785354598, −7.44461818042972989246570578362, −6.60575718879056758640354074663, −5.28722177481878344435313472929, −2.42077868263742505761468144597, 2.77192614700738942165334686396, 4.40022809963226505440061252615, 6.07970326932404353267966841481, 7.75392894977755580774051116033, 9.600630115700918092754364959852, 10.15132886490410513069619768099, 11.47720881076668553188124078781, 12.04724894814315987598064062815, 13.61295598795116149295986137262, 15.03435518990722677381047890312

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