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Properties

Label 1-67-67.59-r0-0-0
Degree $1$
Conductor $67$
Sign $-0.303 + 0.952i$
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  + (−0.959 − 0.281i)2-s + (−0.142 + 0.989i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (0.415 − 0.909i)6-s + (−0.959 − 0.281i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.142 − 0.989i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (−0.654 + 0.755i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−0.142 + 0.989i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (0.415 − 0.909i)6-s + (−0.959 − 0.281i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.142 − 0.989i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (−0.654 + 0.755i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ -0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2993795997 + 0.4097113847i\)
\(L(\frac12)\) \(\approx\) \(0.2993795997 + 0.4097113847i\)
\(L(1)\) \(\approx\) \(0.5442603291 + 0.2826918765i\)
\(L(1)\) \(\approx\) \(0.5442603291 + 0.2826918765i\)

Euler product

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

−32.04234185488104680897516488153, −30.08620601704571422072428934535, −29.29088942504509857183723115011, −28.53580330338420588818962735651, −27.48624645011062570116776285923, −25.89956620655855884794021680736, −25.06803278013610451924426455211, −24.379349622681927840672304525049, −23.217808127304049795933430548461, −21.53306543107906129360198413468, −19.87605834123302046603035060735, −19.34023651948159961986956817893, −18.10594480729008407143228436397, −16.9660122132108381476187123521, −16.30993365518711088955192894522, −14.54735924621063550829723668559, −12.96682467185060621972628353459, −12.10046617217788982895339640406, −10.41187766280811276657977879559, −8.9979166581660409650124953134, −8.091566074182620890339815188564, −6.4997172962851880152527410970, −5.650187377091739226824556120061, −2.59574602408798240037923647609, −0.83788636503762917382257468928, 2.50646560297920871175989619684, 3.88374101485628568403284383183, 6.170424204014966530898791697295, 7.35012074054617165141442188869, 9.45193524249821894978964878830, 9.82783136677854314507151366995, 10.99382365527062840335417171952, 12.28924743829583618212931733321, 14.31636180120684242096283912919, 15.48623734312311415092996014860, 16.68855468850604515681163819530, 17.50335682160346359116472002454, 18.95033085763606355793688460723, 19.90131750993081732241714197059, 21.19989796020735494205839940620, 22.09200580791019419212696809328, 23.175751008909403159604946842833, 25.45251996969344603492150658357, 25.838090070809671622681416158649, 26.92346186010761333644786143474, 27.79278189684680187060055584287, 29.05897341298118827473713858614, 29.69956943654044158972830918181, 31.158368984510365693104102170258, 32.60455933585379733883667318501

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