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Properties

Label 1-1588-1588.1103-r0-0-0
Degree $1$
Conductor $1588$
Sign $-0.961 - 0.275i$
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  + (0.766 − 0.642i)3-s + (−0.342 − 0.939i)5-s + (0.984 − 0.173i)7-s + (0.173 − 0.984i)9-s + (−0.173 − 0.984i)11-s + (0.642 + 0.766i)13-s + (−0.866 − 0.5i)15-s + (−0.866 − 0.5i)17-s + (−0.766 + 0.642i)19-s + (0.642 − 0.766i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.939 − 0.342i)29-s − 31-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.342 − 0.939i)5-s + (0.984 − 0.173i)7-s + (0.173 − 0.984i)9-s + (−0.173 − 0.984i)11-s + (0.642 + 0.766i)13-s + (−0.866 − 0.5i)15-s + (−0.866 − 0.5i)17-s + (−0.766 + 0.642i)19-s + (0.642 − 0.766i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.939 − 0.342i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ -0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2158064954 - 1.534569971i\)
\(L(\frac12)\) \(\approx\) \(0.2158064954 - 1.534569971i\)
\(L(1)\) \(\approx\) \(1.029049034 - 0.6799028716i\)
\(L(1)\) \(\approx\) \(1.029049034 - 0.6799028716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (0.984 - 0.173i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 - T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−20.77610143802003322417947467104, −19.98472514745417712388091026857, −19.67166095745651393777282356588, −18.42098352794572334837309801139, −18.047002559715655545539733452842, −17.24520087816780211137221220036, −16.024880057026291699647228551319, −15.368767314055452076840376736, −14.85963504379930439920948551969, −14.42940922348878460981285383793, −13.37227114223665751017191838846, −12.69964917797361057524008529459, −11.34256872784351152376444823624, −10.96476132221371583634503476327, −10.24662258717083369912837500750, −9.35715443657082030681687674093, −8.420805061811494690023255956493, −7.84290317356520522749088179663, −7.11746955359725989848345483390, −5.99659822130482957059157570736, −4.93710128544507920837187761754, −4.16309629732610243584872060910, −3.45904301206076862541085710571, −2.33677382949271108176789365761, −1.8391673819353732571730617304, 0.46656504599352640252228344417, 1.64715339200933076712928143127, 2.144434012605512940402138554377, 3.68172430062190756267922324653, 4.10417288474648442018219830772, 5.223525853638190549248515068098, 6.15370911105947579722163760736, 7.134709023854220720051390266922, 8.00756839001282730919449761139, 8.63271083016457315147144343011, 8.914649775289718194131046868911, 10.17165291062285856026494493000, 11.37155103081787746720889690156, 11.71326431500197061805994517020, 12.73744529950663360189095686240, 13.49728604144670333624746992430, 13.93390152684258693309623622111, 14.826176682653421543692274962126, 15.60305232651490482836392815768, 16.51333498067405819385425454921, 17.07389365508665444989663773892, 18.25028465313395883721469404179, 18.5442958220802521567978551851, 19.51159888475685883450756048006, 20.167569259442067379507203406504

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