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Properties

Label 1-1617-1617.1007-r1-0-0
Degree $1$
Conductor $1617$
Sign $-0.918 - 0.395i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (−0.0448 + 0.998i)2-s + (−0.995 − 0.0896i)4-s + (0.473 + 0.880i)5-s + (0.134 − 0.990i)8-s + (−0.900 + 0.433i)10-s + (−0.0448 + 0.998i)13-s + (0.983 + 0.178i)16-s + (−0.753 + 0.657i)17-s + (0.309 + 0.951i)19-s + (−0.393 − 0.919i)20-s + (0.222 + 0.974i)23-s + (−0.550 + 0.834i)25-s + (−0.995 − 0.0896i)26-s + (0.858 + 0.512i)29-s + (0.809 − 0.587i)31-s + (−0.222 + 0.974i)32-s + ⋯
L(s)  = 1  + (−0.0448 + 0.998i)2-s + (−0.995 − 0.0896i)4-s + (0.473 + 0.880i)5-s + (0.134 − 0.990i)8-s + (−0.900 + 0.433i)10-s + (−0.0448 + 0.998i)13-s + (0.983 + 0.178i)16-s + (−0.753 + 0.657i)17-s + (0.309 + 0.951i)19-s + (−0.393 − 0.919i)20-s + (0.222 + 0.974i)23-s + (−0.550 + 0.834i)25-s + (−0.995 − 0.0896i)26-s + (0.858 + 0.512i)29-s + (0.809 − 0.587i)31-s + (−0.222 + 0.974i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ -0.918 - 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4226199528 + 2.049715338i\)
\(L(\frac12)\) \(\approx\) \(-0.4226199528 + 2.049715338i\)
\(L(1)\) \(\approx\) \(0.6981680296 + 0.8307873950i\)
\(L(1)\) \(\approx\) \(0.6981680296 + 0.8307873950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0448 + 0.998i)T \)
5 \( 1 + (0.473 + 0.880i)T \)
13 \( 1 + (-0.0448 + 0.998i)T \)
17 \( 1 + (-0.753 + 0.657i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.858 + 0.512i)T \)
41 \( 1 + (-0.134 + 0.990i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.936 + 0.351i)T \)
53 \( 1 + (-0.753 - 0.657i)T \)
59 \( 1 + (0.134 + 0.990i)T \)
61 \( 1 + (0.753 - 0.657i)T \)
67 \( 1 + T \)
71 \( 1 + (0.393 - 0.919i)T \)
73 \( 1 + (0.936 - 0.351i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.0448 + 0.998i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−20.15267401292137674613454600786, −19.276546812351508240914217954318, −18.35949702287614562392340325338, −17.51781560724605867813453750146, −17.35350837647749875980107549046, −16.11902136251790315604603955487, −15.45264557271940430981678961797, −14.22181481671871539889209826557, −13.665778615379530626065951524, −12.87325835078447331757246612775, −12.41999670113993990052329155353, −11.50901567670628866651729617662, −10.70105556520233848116733012261, −9.961739258991981634112554115898, −9.174884510205198163383195045745, −8.59990830104656183877203927439, −7.77818969306579399220121714432, −6.517417292850534095727242994300, −5.42204636298064051561691682249, −4.8202973488427753613342537768, −4.080367633306652945643971626356, −2.76757424887068629062546420896, −2.312644039246332210194213029750, −0.89279739750563201864379351228, −0.53199071205167543548081099050, 1.132681571695821713746995192708, 2.22781247542765090387069420813, 3.43676864874505918742892699382, 4.24067153517810412117275959981, 5.21564859278192762692806785303, 6.23251860459843346297600813886, 6.53819695543857968228052142395, 7.49718496738716457540711469182, 8.21940995991517912027701184625, 9.26045730554069215459721688697, 9.78992987858254685855972295612, 10.66207447689575053162734534633, 11.554836865569805457350032435608, 12.58397357605059253547056389161, 13.56694344718081027138286935285, 13.98266475179136970124333351906, 14.76483817473015166138263210533, 15.37690377990198379327044785971, 16.19790642648940665926625842203, 17.02196765029083423012338003406, 17.612237173477002643940089025035, 18.3363465394641617265575625669, 18.99621392964614520899013354416, 19.60701125691108979351049261234, 20.92559929533128946314010406821

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