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Properties

Label 1-1580-1580.907-r0-0-0
Degree $1$
Conductor $1580$
Sign $0.810 - 0.585i$
Analytic cond. $7.33748$
Root an. cond. $7.33748$
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  + (−0.239 − 0.970i)3-s + (0.239 + 0.970i)7-s + (−0.885 + 0.464i)9-s + (0.748 − 0.663i)11-s + (−0.935 + 0.354i)13-s + (0.935 − 0.354i)17-s + (0.120 − 0.992i)19-s + (0.885 − 0.464i)21-s − i·23-s + (0.663 + 0.748i)27-s + (−0.885 − 0.464i)29-s + (−0.568 + 0.822i)31-s + (−0.822 − 0.568i)33-s + (0.992 + 0.120i)37-s + (0.568 + 0.822i)39-s + ⋯
L(s)  = 1  + (−0.239 − 0.970i)3-s + (0.239 + 0.970i)7-s + (−0.885 + 0.464i)9-s + (0.748 − 0.663i)11-s + (−0.935 + 0.354i)13-s + (0.935 − 0.354i)17-s + (0.120 − 0.992i)19-s + (0.885 − 0.464i)21-s − i·23-s + (0.663 + 0.748i)27-s + (−0.885 − 0.464i)29-s + (−0.568 + 0.822i)31-s + (−0.822 − 0.568i)33-s + (0.992 + 0.120i)37-s + (0.568 + 0.822i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $0.810 - 0.585i$
Analytic conductor: \(7.33748\)
Root analytic conductor: \(7.33748\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (0:\ ),\ 0.810 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340246886 - 0.4334069294i\)
\(L(\frac12)\) \(\approx\) \(1.340246886 - 0.4334069294i\)
\(L(1)\) \(\approx\) \(0.9977459373 - 0.2207103175i\)
\(L(1)\) \(\approx\) \(0.9977459373 - 0.2207103175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.239 - 0.970i)T \)
7 \( 1 + (0.239 + 0.970i)T \)
11 \( 1 + (0.748 - 0.663i)T \)
13 \( 1 + (-0.935 + 0.354i)T \)
17 \( 1 + (0.935 - 0.354i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (-0.568 + 0.822i)T \)
37 \( 1 + (0.992 + 0.120i)T \)
41 \( 1 + (-0.748 - 0.663i)T \)
43 \( 1 + (0.663 - 0.748i)T \)
47 \( 1 + (0.992 - 0.120i)T \)
53 \( 1 + (-0.239 + 0.970i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (0.120 - 0.992i)T \)
67 \( 1 + (0.822 - 0.568i)T \)
71 \( 1 + (0.970 + 0.239i)T \)
73 \( 1 + (0.935 + 0.354i)T \)
83 \( 1 + (0.935 - 0.354i)T \)
89 \( 1 + (0.970 - 0.239i)T \)
97 \( 1 + (0.992 + 0.120i)T \)
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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

−20.37940342269953102105850402887, −20.23614119494600852130422884500, −19.236693450137820945510169836280, −18.19982007877131678906029099270, −17.33158161414129749686362790080, −16.69845752915385571510569870063, −16.470336399226794614354473480298, −15.12787868816148235735640394728, −14.589275718029002437049204205628, −14.231916637516430248974295758738, −12.87545733572620734224871399680, −12.21489443388155153308169389631, −11.33415738160142730936149784474, −10.56981080762739973666554386374, −9.8591695578860987026639953788, −9.43709760580094395969479466898, −8.174367576996235063234649904044, −7.51773380595640761912469239141, −6.50982263317070099313265929288, −5.58913717875492149852457310392, −4.718694166492384808153870516252, −4.02509248042118854648289416573, −3.34581604925995043851541877079, −2.05522700635782052317875786678, −0.796805221833943134704930362801, 0.78559880427518288145410623934, 1.86515383607062738719439310384, 2.62709028330832514542915860384, 3.60019614391763965950137595935, 5.0420645423933029138032792641, 5.563478113648026471162925663243, 6.41952235586084049967081582662, 7.29987658649576555854747640449, 7.90930051297383454732389939606, 9.03417898384665684137659052110, 9.3569475700135597981360930967, 10.78327838467092069869100000666, 11.62456083227875668459313372789, 11.98854113348628931854943581905, 12.73717186747966883278325348626, 13.726800268729105354674207698945, 14.2675005394159022046599026259, 15.08728132968722433950644662667, 15.98753414623944040498686609776, 17.06515379311738392090877193067, 17.28081940906482023520906374394, 18.46416242744911632131141097125, 18.76687489601777721173226616152, 19.569122297280468915634833196317, 20.16395226160879856712545443956

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