Properties

Label 1-667-667.157-r0-0-0
Degree $1$
Conductor $667$
Sign $0.0819 + 0.996i$
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  + (−0.755 − 0.654i)2-s + (0.540 + 0.841i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (0.959 − 0.281i)7-s + (0.540 − 0.841i)8-s + (−0.415 + 0.909i)9-s + (0.281 − 0.959i)10-s + (−0.755 + 0.654i)11-s + (−0.755 + 0.654i)12-s + (0.959 + 0.281i)13-s + (−0.909 − 0.415i)14-s + (−0.540 + 0.841i)15-s + (−0.959 + 0.281i)16-s + (0.989 + 0.142i)17-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (0.540 + 0.841i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (0.959 − 0.281i)7-s + (0.540 − 0.841i)8-s + (−0.415 + 0.909i)9-s + (0.281 − 0.959i)10-s + (−0.755 + 0.654i)11-s + (−0.755 + 0.654i)12-s + (0.959 + 0.281i)13-s + (−0.909 − 0.415i)14-s + (−0.540 + 0.841i)15-s + (−0.959 + 0.281i)16-s + (0.989 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0819 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0819 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.0819 + 0.996i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.0819 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9870031702 + 0.9091569914i\)
\(L(\frac12)\) \(\approx\) \(0.9870031702 + 0.9091569914i\)
\(L(1)\) \(\approx\) \(0.9824363307 + 0.3238827201i\)
\(L(1)\) \(\approx\) \(0.9824363307 + 0.3238827201i\)

Euler product

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

−23.21469580951098980415364264160, −21.329248667574887653939827421251, −20.84920716555099268615715836621, −20.05626117882364675092230689266, −19.08265664695728291358104986909, −18.32524898921016397021348636418, −17.859191812524136636105373727529, −16.9168183369827128550442382899, −16.15756922256613137143806163730, −15.10424460686506165118267475371, −14.36802517473109321202785033095, −13.49757291148857219097168720536, −12.81475276217498104905917902869, −11.59016721501784827328918660679, −10.72212112233345369818376685388, −9.485656385406576370948573500107, −8.667743615469465203253024540, −8.13506545386127772471402117029, −7.53595438704848417736199003034, −5.98422757578736263092497812753, −5.73329957125177618805901123875, −4.421586394575647460747661007609, −2.66914907335436209287233926233, −1.62092598275826465145818794181, −0.82941012969308328708024548070, 1.64733796043396115877275923758, 2.41963435744375441865875756822, 3.470957445491784709538572317116, 4.27855744894619631127689804824, 5.50406310897121630242265600271, 6.97912553000638186870052078952, 7.89644137613795026949073079249, 8.56373267439556824721300259948, 9.616240502381381798192440667907, 10.50462782395582199997855575462, 10.72841023776329965508570568222, 11.70667463741280134127196276745, 13.01301858092881240592565853106, 13.91469273361802349491794104971, 14.72260897023252548918841995055, 15.5149825661910145796613392029, 16.56328276867922264642240333188, 17.334553033546137357535535245829, 18.26526372462631427561490314553, 18.75921767751761136576323721483, 19.83246846663934897899330134386, 20.65783151333890188192372528246, 21.28313566096116974854387342233, 21.609266371115014912033816133867, 22.82074360383724429668126779157

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