Properties

Label 1-667-667.447-r0-0-0
Degree $1$
Conductor $667$
Sign $0.707 - 0.706i$
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  + (−0.989 − 0.142i)2-s + (−0.909 + 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (−0.841 − 0.540i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (0.540 + 0.841i)10-s + (−0.989 + 0.142i)11-s + (−0.989 + 0.142i)12-s + (−0.841 + 0.540i)13-s + (0.755 + 0.654i)14-s + (0.909 + 0.415i)15-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (−0.909 + 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (−0.841 − 0.540i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (0.540 + 0.841i)10-s + (−0.989 + 0.142i)11-s + (−0.989 + 0.142i)12-s + (−0.841 + 0.540i)13-s + (0.755 + 0.654i)14-s + (0.909 + 0.415i)15-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.707 - 0.706i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.707 - 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2372102608 - 0.09810933347i\)
\(L(\frac12)\) \(\approx\) \(0.2372102608 - 0.09810933347i\)
\(L(1)\) \(\approx\) \(0.3555397324 + 0.02230985298i\)
\(L(1)\) \(\approx\) \(0.3555397324 + 0.02230985298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 + (-0.909 + 0.415i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (-0.989 + 0.142i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.281 + 0.959i)T \)
31 \( 1 + (0.909 + 0.415i)T \)
37 \( 1 + (-0.755 - 0.654i)T \)
41 \( 1 + (-0.755 + 0.654i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (-0.909 - 0.415i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (-0.540 - 0.841i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (0.909 - 0.415i)T \)
97 \( 1 + (0.755 - 0.654i)T \)
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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

−22.84441262401963357742779700305, −22.319826205727727875229523395989, −21.29687036301266307678904131922, −20.110170158773844671381719442639, −19.0864115785229735234703117183, −18.9091403336662960191896746043, −17.997713374499542621785227990496, −17.347507030140595463586998261913, −16.25558218377619425269461654173, −15.68097191757565842191640139908, −15.11693137224388111235846980285, −13.64238568186863548443266253264, −12.45812382173435942443263056842, −11.8992682277672010916355727801, −10.9686598900235092093391660028, −10.31282674051431537077147980038, −9.49257164612885728527485306843, −8.1541576236399124529405040039, −7.3886397781821678831043491676, −6.752641100370302896482924742907, −5.85301521251670360712481890534, −4.881865051267814162607868968660, −2.96664710120963908544678461857, −2.44982458046002292675829082559, −0.58028907815652984236447489885, 0.37135369016832128861597089795, 1.70096874891928178055788726536, 3.32084354324618596157660617433, 4.22724835073983555865708956898, 5.37911001358351796381733241975, 6.43514429790088845583725613471, 7.3286635277235180666486126346, 8.18640841287269243457045138153, 9.27115997549118320313227084215, 10.16783235776406760134141710635, 10.56195657455516009802050375137, 11.77499683222498738657734307023, 12.37932533480784734401567145955, 13.02322514519529374785451158975, 14.791755521609807817710142902039, 15.79262449599854911023283328113, 16.1875614425915693727243613520, 17.00552162453046642873244997091, 17.42655545440360524629097731079, 18.74617870020247715748667700867, 19.250517406046111103427678204021, 20.18881731060192124263017164057, 20.967109128182548686072137328497, 21.62833372556638506370525840317, 22.83187007799732520244753950280

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