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Properties

Label 1-67-67.64-r0-0-0
Degree $1$
Conductor $67$
Sign $0.0773 + 0.997i$
Analytic cond. $0.311146$
Root an. cond. $0.311146$
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  + (0.841 + 0.540i)2-s + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 − 0.989i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)17-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 − 0.989i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0773 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0773 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.0773 + 0.997i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.0773 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7629228450 + 0.7060476343i\)
\(L(\frac12)\) \(\approx\) \(0.7629228450 + 0.7060476343i\)
\(L(1)\) \(\approx\) \(1.014451657 + 0.5314353995i\)
\(L(1)\) \(\approx\) \(1.014451657 + 0.5314353995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.959 - 0.281i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + T \)
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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

−31.74321786525679383837461461271, −30.71220698803436206196306049067, −29.49797146921689141211253614807, −28.590092536541131530847194899247, −27.72151307169010229166293962669, −26.689230053463492995372759896604, −24.28088847386893432796787769463, −23.91600441646009354919197510738, −23.03288408943608549136291032237, −21.55089826792661287820622925352, −20.971463901243918845792339228302, −19.67770794130199995180486917847, −18.32492090490889519985611064279, −16.69092383871735544522684613458, −15.90233471831847773383584933933, −14.459752102190001943604782453447, −13.06825290412227969721940963997, −11.83070095645719680893361147141, −11.1991584699270425535702805505, −9.87996725670698652632119478358, −7.82698509941230163179353287809, −6.001777977219258589826835355070, −4.80160364296527433802522289728, −3.89011457254991300141644975972, −1.275049033256970260999779443531, 2.67161347791208181140911212010, 4.63300632032539017988704011150, 5.61500334723326465217175588682, 7.149479512134417585814445696, 7.91050711654474788088693609429, 10.52960897400053523623386532578, 11.72423767098485407139486636482, 12.42372118681701609189725225975, 14.01897290411507577553850054701, 15.31659049331410424779502665921, 16.01397506606287851630723905111, 17.75578596922988421800079667193, 18.2196477212338562952806809812, 20.20067253041817431250862625736, 21.610679516487719216214610894476, 22.57785648916085293816955584731, 23.296805352479392520054996941756, 24.291298213782162325754770716631, 25.3485826004381647498675220990, 26.83748379656782448176186366509, 27.837526375885817030255972143890, 29.25311526913555245994187826716, 30.510309224058408242979785677102, 30.87234829963741890266540094052, 32.33833163737591062593522252971

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