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Properties

Label 1-67-67.51-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.465 - 0.884i$
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  + (−0.928 − 0.371i)2-s + (0.654 − 0.755i)3-s + (0.723 + 0.690i)4-s + (−0.841 + 0.540i)5-s + (−0.888 + 0.458i)6-s + (0.786 − 0.618i)7-s + (−0.415 − 0.909i)8-s + (−0.142 − 0.989i)9-s + (0.981 − 0.189i)10-s + (0.888 + 0.458i)11-s + (0.995 − 0.0950i)12-s + (−0.580 − 0.814i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.0475 + 0.998i)16-s + (0.723 − 0.690i)17-s + ⋯
L(s)  = 1  + (−0.928 − 0.371i)2-s + (0.654 − 0.755i)3-s + (0.723 + 0.690i)4-s + (−0.841 + 0.540i)5-s + (−0.888 + 0.458i)6-s + (0.786 − 0.618i)7-s + (−0.415 − 0.909i)8-s + (−0.142 − 0.989i)9-s + (0.981 − 0.189i)10-s + (0.888 + 0.458i)11-s + (0.995 − 0.0950i)12-s + (−0.580 − 0.814i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.0475 + 0.998i)16-s + (0.723 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.465 - 0.884i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.465 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5610476145 - 0.9295192667i\)
\(L(\frac12)\) \(\approx\) \(0.5610476145 - 0.9295192667i\)
\(L(1)\) \(\approx\) \(0.7252745480 - 0.4265363599i\)
\(L(1)\) \(\approx\) \(0.7252745480 - 0.4265363599i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.928 - 0.371i)T \)
3 \( 1 + (0.654 - 0.755i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (0.786 - 0.618i)T \)
11 \( 1 + (0.888 + 0.458i)T \)
13 \( 1 + (-0.580 - 0.814i)T \)
17 \( 1 + (0.723 - 0.690i)T \)
19 \( 1 + (-0.786 - 0.618i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.580 + 0.814i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.235 - 0.971i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.981 + 0.189i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
71 \( 1 + (0.723 + 0.690i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (0.0475 + 0.998i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
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

−32.09328398339832467446057869264, −31.32957933735483450815838509734, −29.860880831267664776334348550081, −28.13492721223593510909139763098, −27.61905370572441595477739163150, −26.86663659965503113378868262921, −25.61163603215774297066664546217, −24.60184164568805518199810777109, −23.70636438738249480024658948276, −21.755146182575494648323140232048, −20.727301896500405852592696732757, −19.5629513661362795692961856673, −18.9132597949400911841120788251, −17.06493000648547477911897017220, −16.303027680290164817918205414017, −15.04408819537026068290329289648, −14.415280368711310766585683819050, −12.01678942869342707480120132312, −10.95530991337511658445257893609, −9.3589570583898969095614957333, −8.57423313786400086140873645892, −7.58426412831882347840407831489, −5.50871039635781330554736678444, −3.93417930761515235183729574590, −1.79908957334904189853951495431, 0.72529800243408573350276801595, 2.43375298507177956356641652321, 3.904558514586871407406272960810, 6.903932221329547815503413093694, 7.60975833590611258321863533021, 8.67201654228346442835457226020, 10.2546630712627529311113033600, 11.58224878675793990838914846151, 12.507714542568628918817249857062, 14.30093181465422821837577692349, 15.28008255159135450114175215603, 17.05111999769145553922965143039, 18.0122215349115256593991455977, 19.10494782775015569245598423704, 19.95358649157850374015011741883, 20.72568104318138695942088963572, 22.528594497904429513827203787001, 23.93120609561471523865612030685, 24.95506071733374285842586851771, 26.04452518065133350680738279005, 27.09840200499377036593957008748, 27.762406096917210986847244225304, 29.5283643599863572308340202933, 30.291325356695603839842265219143, 30.78823535000466228957224868520

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