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Properties

Label 1-667-667.337-r0-0-0
Degree $1$
Conductor $667$
Sign $0.803 - 0.594i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.925 − 0.377i)2-s + (−0.470 + 0.882i)3-s + (0.714 − 0.699i)4-s + (−0.862 + 0.505i)5-s + (−0.101 + 0.994i)6-s + (−0.794 + 0.607i)7-s + (0.396 − 0.917i)8-s + (−0.557 − 0.830i)9-s + (−0.607 + 0.794i)10-s + (0.162 − 0.986i)11-s + (0.281 + 0.959i)12-s + (0.768 − 0.639i)13-s + (−0.505 + 0.862i)14-s + (−0.0407 − 0.999i)15-s + (0.0203 − 0.999i)16-s + (−0.540 − 0.841i)17-s + ⋯
L(s)  = 1  + (0.925 − 0.377i)2-s + (−0.470 + 0.882i)3-s + (0.714 − 0.699i)4-s + (−0.862 + 0.505i)5-s + (−0.101 + 0.994i)6-s + (−0.794 + 0.607i)7-s + (0.396 − 0.917i)8-s + (−0.557 − 0.830i)9-s + (−0.607 + 0.794i)10-s + (0.162 − 0.986i)11-s + (0.281 + 0.959i)12-s + (0.768 − 0.639i)13-s + (−0.505 + 0.862i)14-s + (−0.0407 − 0.999i)15-s + (0.0203 − 0.999i)16-s + (−0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.803 - 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.505077339 - 0.4961345852i\)
\(L(\frac12)\) \(\approx\) \(1.505077339 - 0.4961345852i\)
\(L(1)\) \(\approx\) \(1.284096782 - 0.09906382583i\)
\(L(1)\) \(\approx\) \(1.284096782 - 0.09906382583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.925 - 0.377i)T \)
3 \( 1 + (-0.470 + 0.882i)T \)
5 \( 1 + (-0.862 + 0.505i)T \)
7 \( 1 + (-0.794 + 0.607i)T \)
11 \( 1 + (0.162 - 0.986i)T \)
13 \( 1 + (0.768 - 0.639i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.699 + 0.714i)T \)
31 \( 1 + (0.983 + 0.182i)T \)
37 \( 1 + (-0.830 + 0.557i)T \)
41 \( 1 + (0.989 + 0.142i)T \)
43 \( 1 + (0.983 - 0.182i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.452 - 0.891i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (0.806 + 0.591i)T \)
67 \( 1 + (0.986 - 0.162i)T \)
71 \( 1 + (-0.742 - 0.670i)T \)
73 \( 1 + (-0.320 - 0.947i)T \)
79 \( 1 + (0.999 - 0.0203i)T \)
83 \( 1 + (-0.996 - 0.0815i)T \)
89 \( 1 + (-0.0407 + 0.999i)T \)
97 \( 1 + (0.359 - 0.933i)T \)
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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

−22.86147043522954879614297154957, −22.63334180586404092975484092272, −21.37247521942716811676785201109, −20.24648749410168157305206934213, −19.77851845030095995669013053888, −18.97654849324119549773066608699, −17.61067039684631053318238750695, −17.06562542395534134284473514544, −16.0720178654743260771370926860, −15.658762430414522767756407202879, −14.42688141123220630342263179685, −13.44584911102132289997917713253, −12.93498545554189848880274750482, −12.178773225943217412687897268255, −11.478345890987494032071311370498, −10.60496118711293473690575878248, −9.01105208701096781995481572804, −7.98591338522847166558421670721, −7.09629691710961061395121668001, −6.66681053838845196077194165244, −5.58209641568324251102155652004, −4.449220515951418211093542221137, −3.82044167928003102832253221300, −2.48022753736778734242586437998, −1.16912576196818184445258665179, 0.70213289235799885919967684684, 2.84499068549408255414588371921, 3.32004138404408181572496536971, 4.1279574984703189216720870961, 5.29232189272130739523032475213, 6.05418269172499125007049513019, 6.79171980029397605813943538272, 8.25008124355534493538938817706, 9.388192782737784234184885028877, 10.33015834543733161386608456222, 11.11640669699682228518379558453, 11.708007633803808288231613583152, 12.45112155582872919922226064626, 13.601950950719769412816774547154, 14.460921243706763668167770029624, 15.41229476039973425532116997324, 15.99177340336274373831675868597, 16.26987393391116641832316711676, 17.91642828703074908121693759863, 18.87523438026797628965424547253, 19.52966224726327449601099517793, 20.52307240154592808055495090369, 21.13475870847499120323965256466, 22.303397459378049177573510863339, 22.46614914067491611256602688076

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