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Properties

Label 1-1588-1588.1215-r0-0-0
Degree $1$
Conductor $1588$
Sign $-0.0934 - 0.995i$
Analytic cond. $7.37464$
Root an. cond. $7.37464$
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  + (0.356 + 0.934i)3-s + (−0.312 + 0.950i)5-s + (−0.666 + 0.745i)7-s + (−0.745 + 0.666i)9-s + (−0.902 − 0.429i)11-s + (−0.592 + 0.805i)13-s + (−0.998 + 0.0475i)15-s + (−0.371 + 0.928i)17-s + (0.975 − 0.220i)19-s + (−0.934 − 0.356i)21-s + (−0.823 + 0.567i)23-s + (−0.805 − 0.592i)25-s + (−0.888 − 0.458i)27-s + (−0.999 + 0.0317i)29-s + (0.654 − 0.755i)31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s + (−0.312 + 0.950i)5-s + (−0.666 + 0.745i)7-s + (−0.745 + 0.666i)9-s + (−0.902 − 0.429i)11-s + (−0.592 + 0.805i)13-s + (−0.998 + 0.0475i)15-s + (−0.371 + 0.928i)17-s + (0.975 − 0.220i)19-s + (−0.934 − 0.356i)21-s + (−0.823 + 0.567i)23-s + (−0.805 − 0.592i)25-s + (−0.888 − 0.458i)27-s + (−0.999 + 0.0317i)29-s + (0.654 − 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0934 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0934 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $-0.0934 - 0.995i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ -0.0934 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3196190175 + 0.3510082797i\)
\(L(\frac12)\) \(\approx\) \(-0.3196190175 + 0.3510082797i\)
\(L(1)\) \(\approx\) \(0.5854368455 + 0.5234583192i\)
\(L(1)\) \(\approx\) \(0.5854368455 + 0.5234583192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (0.356 + 0.934i)T \)
5 \( 1 + (-0.312 + 0.950i)T \)
7 \( 1 + (-0.666 + 0.745i)T \)
11 \( 1 + (-0.902 - 0.429i)T \)
13 \( 1 + (-0.592 + 0.805i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (0.975 - 0.220i)T \)
23 \( 1 + (-0.823 + 0.567i)T \)
29 \( 1 + (-0.999 + 0.0317i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (-0.0158 + 0.999i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (-0.997 + 0.0634i)T \)
53 \( 1 + (0.998 + 0.0475i)T \)
59 \( 1 + (-0.158 - 0.987i)T \)
61 \( 1 + (0.592 + 0.805i)T \)
67 \( 1 + (0.857 + 0.513i)T \)
71 \( 1 + (-0.189 + 0.981i)T \)
73 \( 1 + (-0.386 + 0.922i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (-0.993 + 0.110i)T \)
97 \( 1 + (0.527 - 0.849i)T \)
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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

−19.83349311765769356933189615120, −19.531909201164350932184183611305, −18.315047665863713818877167586609, −17.87127231559618644602891100615, −17.023126841551478727838239442457, −16.13583165779459875387147661783, −15.65895009841599302780724410633, −14.5115202839418759353865558062, −13.72574939844799501564179836844, −13.05820232876013739983665180852, −12.54584965950124588955471941621, −11.91734840587620894025668892209, −10.82819493908204456400956870190, −9.7945447250202167203234079214, −9.23092295402067659775856562341, −8.10700790451460135066943480434, −7.59974931952312897325281677306, −7.01951195526550948864960305337, −5.8306095866360977821654584757, −5.08671399374755250114608161218, −4.05009036556578801601656043293, −3.05108566467594716257876464481, −2.23601329333778782680061026358, −0.97729237051508563597802683077, −0.18532364110533657141764495463, 2.15902057260385533417583617214, 2.75987111423273537792907325808, 3.55750682711957383319506355397, 4.33474535152938014292528173421, 5.50790304652787885139692883440, 6.07693068470576538560787070486, 7.20287863262079903268430169544, 8.019172732385359722264820681667, 8.84026596742451862091550279216, 9.841892305371262933991243921998, 10.08291209056334237238057166685, 11.31157766020518238330696863537, 11.56982986382128515455306794276, 12.82846591135183320139347624374, 13.6889606721732902225979872346, 14.43660433380303990654622665471, 15.18521135585213841484354125386, 15.72318446674683725783696085211, 16.25540381713165548359498659735, 17.251651423232021949977790951026, 18.21080968539431452254405763608, 19.01772687520126837685136636550, 19.399493843765866183547973947613, 20.27638353900775362145334395923, 21.2248990835019600555598403915

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