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Properties

Label 1-67-67.49-r0-0-0
Degree $1$
Conductor $67$
Sign $0.884 + 0.466i$
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  + (−0.327 − 0.945i)2-s + (0.415 + 0.909i)3-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (0.723 − 0.690i)6-s + (0.981 + 0.189i)7-s + (0.841 + 0.540i)8-s + (−0.654 + 0.755i)9-s + (0.580 + 0.814i)10-s + (0.723 + 0.690i)11-s + (−0.888 − 0.458i)12-s + (0.0475 + 0.998i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (0.235 − 0.971i)16-s + (−0.786 − 0.618i)17-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (0.415 + 0.909i)3-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (0.723 − 0.690i)6-s + (0.981 + 0.189i)7-s + (0.841 + 0.540i)8-s + (−0.654 + 0.755i)9-s + (0.580 + 0.814i)10-s + (0.723 + 0.690i)11-s + (−0.888 − 0.458i)12-s + (0.0475 + 0.998i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (0.235 − 0.971i)16-s + (−0.786 − 0.618i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7492964729 + 0.1855687512i\)
\(L(\frac12)\) \(\approx\) \(0.7492964729 + 0.1855687512i\)
\(L(1)\) \(\approx\) \(0.8679368596 + 0.04745357172i\)
\(L(1)\) \(\approx\) \(0.8679368596 + 0.04745357172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.327 - 0.945i)T \)
3 \( 1 + (0.415 + 0.909i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (0.981 + 0.189i)T \)
11 \( 1 + (0.723 + 0.690i)T \)
13 \( 1 + (0.0475 + 0.998i)T \)
17 \( 1 + (-0.786 - 0.618i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
23 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.0475 - 0.998i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.928 - 0.371i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.580 - 0.814i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.786 + 0.618i)T \)
73 \( 1 + (0.723 - 0.690i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (0.235 - 0.971i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
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

−31.91180145287161943687017149310, −30.98404049496636086906507140131, −30.03435867599763825455200319921, −28.35715728845352307740099403615, −27.288054611172430946131287434062, −26.48847998507363446992575999382, −25.00771348670756018758762659672, −24.28132924695124794978654434831, −23.64103166219974861653850395100, −22.35666136593634562160753361710, −20.24210935441086887925905114445, −19.48575700057573767401990726051, −18.22289108831875520451155591975, −17.37926303479539347700597782434, −15.95114720897401618100692643987, −14.74762793533473119096764898239, −13.85162479774743859882121880286, −12.42098114260366220419814346816, −10.99518796060256176299056814761, −8.850189117287163057577608172639, −8.08532610699734307881781029376, −7.162561215364008887270241104565, −5.59053572189227000832371528448, −3.841669938420712557298148518155, −1.16590467472096318942326007714, 2.22679579646755516324469845772, 3.91391010053088313586512603635, 4.674471022641845054633186923835, 7.56304947628579913487622443699, 8.74382958658421988139989894188, 9.81583390770055089072448434894, 11.37859362603248039669127172114, 11.7221656927878087776155340966, 13.84752050474167424910229471070, 14.85593208080427502794458010296, 16.18939253699611073369642725726, 17.57746501570574829763625332973, 18.876009220735533232926290405563, 20.03335999326080824786445752801, 20.67438670421583788399339048724, 21.971999294270903012792891492328, 22.73025261994922652641989991148, 24.373712540302355235373596821761, 26.1147300693187139120954056606, 26.851165978601509017305126807920, 27.739587509479179799422993459317, 28.41416688227316714758633361483, 30.19978082461648686850430451590, 31.08992670347259290615659363115, 31.553926166091846326295322124004

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