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Properties

Label 1-1575-1575.1208-r0-0-0
Degree $1$
Conductor $1575$
Sign $0.788 + 0.615i$
Analytic cond. $7.31426$
Root an. cond. $7.31426$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)8-s + (0.104 + 0.994i)11-s + (0.406 + 0.913i)13-s + (−0.809 − 0.587i)16-s + (0.743 + 0.669i)17-s + (0.978 + 0.207i)19-s + (0.743 − 0.669i)22-s + (0.406 − 0.913i)23-s + (0.5 − 0.866i)26-s + (−0.978 + 0.207i)29-s + (0.309 + 0.951i)31-s + i·32-s + (0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)8-s + (0.104 + 0.994i)11-s + (0.406 + 0.913i)13-s + (−0.809 − 0.587i)16-s + (0.743 + 0.669i)17-s + (0.978 + 0.207i)19-s + (0.743 − 0.669i)22-s + (0.406 − 0.913i)23-s + (0.5 − 0.866i)26-s + (−0.978 + 0.207i)29-s + (0.309 + 0.951i)31-s + i·32-s + (0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ 0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020819090 + 0.3511860194i\)
\(L(\frac12)\) \(\approx\) \(1.020819090 + 0.3511860194i\)
\(L(1)\) \(\approx\) \(0.8285571054 - 0.06757504490i\)
\(L(1)\) \(\approx\) \(0.8285571054 - 0.06757504490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.406 - 0.913i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.207 + 0.978i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.951 - 0.309i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−20.34110589166867677970788491900, −19.49716516345303706821084335070, −18.68955471871330461783167798960, −18.30442037819970212356191375559, −17.294888403347745439988535691118, −16.786057055245105596637246373146, −15.89590717291508796204830944019, −15.43160103932361063551283364386, −14.555091572769759455565802693538, −13.62643178520324456980110346717, −13.336415786795458349209891383888, −11.90369922769222716741279269057, −11.18509119508353194933702081888, −10.36589677183082319497237077470, −9.52646710043313593661598977795, −8.904298030564362472982394739433, −7.88863560657981401481076192553, −7.51260774062530377111475553725, −6.40577296273157674548390996235, −5.58839299207018129939179520665, −5.13839638038928183800976686310, −3.75650658654003938317544923953, −2.898523607018708563736585651086, −1.429018075707417824391434830332, −0.57111677216511484965433337507, 1.18479341424284915099043189562, 1.87287301530004659497848174295, 2.92612382398488042937754071393, 3.85818792412441768604749600181, 4.5609238816612560292862127968, 5.66512269595868797843219856396, 6.93715346354852537013327572843, 7.45674329772313645110235232282, 8.50154332114189746774192672805, 9.16818666807713965486476267179, 9.92042369590629817180960718620, 10.63564687322219935052483085973, 11.41972001329442736905811228094, 12.33870822399182778324108360578, 12.61683727281212132793586640242, 13.80291689867722921006984264184, 14.37671062215363262019905593173, 15.47950240573137531217100827216, 16.34864656514500688785623095666, 17.00967487284422267951170890134, 17.67768076912535418293847326998, 18.65981551951359601728708567734, 18.86106558089144366840455554737, 20.01412832691333359659843184742, 20.41395270811447322119984824316

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