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Properties

Label 1-67-67.45-r1-0-0
Degree $1$
Conductor $67$
Sign $0.455 + 0.890i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.455 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369629317 + 0.8376463815i\)
\(L(\frac12)\) \(\approx\) \(1.369629317 + 0.8376463815i\)
\(L(1)\) \(\approx\) \(1.063232802 + 0.3679573203i\)
\(L(1)\) \(\approx\) \(1.063232802 + 0.3679573203i\)

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

−31.59738618834488549407400417931, −30.207362731825202059872977748731, −29.32118910740067237599383244506, −28.31005012253305362886951077238, −27.0135354339952858286590898906, −26.20072803986800816475524668981, −25.267401761481502331721946176462, −24.30940212456184537244247085736, −22.11570063505672744407846444110, −21.20452132978188128804676049241, −20.16509010923934783183278395325, −19.46634438073434232144064805640, −18.22569059147395980921757318087, −16.53723918024886531452560791078, −16.203208505768633499143494968014, −13.984260413990807351900781359122, −13.19433612770725507221100238741, −11.65959403220018419189921617405, −9.85403419803377776487027651945, −9.401632053809930023990079861934, −8.20632274005064085297210176401, −6.65238324211628244313151192729, −4.19389158159168737656219989232, −2.8001869815507215905817458218, −1.11213353390489316654576811079, 1.71271141402984651607222317689, 3.163944411479668328292701579668, 5.905277258695470374016831192888, 6.94827610406545022008786722861, 8.23302588550341353440006612856, 9.60092274802826390806516881558, 10.20548597142560955349836267785, 12.35885698415604089323787688383, 13.92105181215033523981360535707, 14.88495003957211474923834717920, 15.81902846070944133083293086876, 17.53521090274154772533362298393, 18.39871579573901798522031793921, 19.397797782536908294084838026016, 20.31729700077782076216896356666, 21.93173350239395427114623977815, 23.24803414945895139140024607750, 24.93869030513333261329575734465, 25.368722026209607383975156594930, 26.144012198886595400832644528211, 27.27499453791844828610910752162, 28.606045176923831917018275411672, 29.732862793376383446901037698879, 30.68655337467572371856212629536, 32.360855609139776713144466189678

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