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Properties

Label 1-67-67.23-r0-0-0
Degree $1$
Conductor $67$
Sign $0.865 + 0.500i$
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  + (−0.888 + 0.458i)2-s + (−0.959 − 0.281i)3-s + (0.580 − 0.814i)4-s + (−0.654 + 0.755i)5-s + (0.981 − 0.189i)6-s + (0.0475 − 0.998i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (0.235 − 0.971i)10-s + (0.981 + 0.189i)11-s + (−0.786 + 0.618i)12-s + (0.928 + 0.371i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (−0.327 − 0.945i)16-s + (0.580 + 0.814i)17-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (−0.959 − 0.281i)3-s + (0.580 − 0.814i)4-s + (−0.654 + 0.755i)5-s + (0.981 − 0.189i)6-s + (0.0475 − 0.998i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (0.235 − 0.971i)10-s + (0.981 + 0.189i)11-s + (−0.786 + 0.618i)12-s + (0.928 + 0.371i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (−0.327 − 0.945i)16-s + (0.580 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4476095019 + 0.1201365145i\)
\(L(\frac12)\) \(\approx\) \(0.4476095019 + 0.1201365145i\)
\(L(1)\) \(\approx\) \(0.5423510219 + 0.09358056119i\)
\(L(1)\) \(\approx\) \(0.5423510219 + 0.09358056119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.888 + 0.458i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (0.928 + 0.371i)T \)
17 \( 1 + (0.580 + 0.814i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.995 + 0.0950i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.235 + 0.971i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (0.981 - 0.189i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−31.8957994119506463419416819895, −30.5735969254018922718377552739, −29.43494590962517682776110222545, −28.19529506104407408670767129923, −27.89556659030022287831871655033, −26.98367854189166641770042641485, −25.32299983148287022394431876389, −24.36721221923527848961241317978, −22.91734205866315282571446456235, −21.709237725141650891684361319293, −20.75998473445905613113211797305, −19.45097742329842484192601704812, −18.34454714863245805110219352466, −17.26505513282326240584527461009, −16.17962746287633893007582454096, −15.45243877598710280508225033382, −12.852064443975353622090878802373, −11.79903084222927874260656321702, −11.20969354155861326455035451786, −9.475336159279833181076349694346, −8.59288972043066946275005767119, −6.90074790911222683352880692616, −5.267459972123692533635767224641, −3.54297348762982222466817064650, −1.11083343202195577276579051098, 1.28467241471813464839178007330, 4.069593453637532472664190925779, 6.15590436368281500966343579279, 6.967901683426904726580625417299, 8.11418599937185890644803537308, 10.09746421134974055212241306893, 10.945087430718118045569730372, 11.957715963136077650821040303745, 13.972427176011718566723256008958, 15.26876944860360647933959989055, 16.6112207821727810678268778395, 17.22339245962497513838374276470, 18.61638383297541808178923000242, 19.24149528885590259979516048654, 20.72031467741408827287106091475, 22.65287023591780614982027866014, 23.306475320072030770457017182745, 24.27749471205526149330704842719, 25.677675334745769871246666288095, 26.860472260773799329469822923331, 27.56319788401148992227999095951, 28.62915883510868055050324641601, 29.88314797607390580825018098995, 30.498938968312542280240251636219, 32.67546744738897062913095880002

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