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Properties

Label 1-1925-1925.1187-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.653 + 0.756i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.207 + 0.978i)2-s + (0.406 − 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (−0.743 + 0.669i)12-s + (0.951 + 0.309i)13-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (0.866 − 0.5i)18-s + (0.913 − 0.406i)19-s + (−0.207 + 0.978i)23-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (0.406 − 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (−0.743 + 0.669i)12-s + (0.951 + 0.309i)13-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (0.866 − 0.5i)18-s + (0.913 − 0.406i)19-s + (−0.207 + 0.978i)23-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.653 + 0.756i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402123091 + 0.6414813622i\)
\(L(\frac12)\) \(\approx\) \(1.402123091 + 0.6414813622i\)
\(L(1)\) \(\approx\) \(1.046154899 + 0.2286773399i\)
\(L(1)\) \(\approx\) \(1.046154899 + 0.2286773399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.994 + 0.104i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.207 + 0.978i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.743 + 0.669i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.743 + 0.669i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−20.31180389036692064353671406124, −19.17087069489225283590113064843, −18.74527545012170276917881315201, −17.91460034681780890120824375285, −16.94668276735977687705977857522, −16.418415636056306179092207402, −15.55721635623299971686229317146, −14.67681410546498022606728511528, −13.91745388337699812115818337989, −13.424197183105903921072726038874, −12.341247142809319267038898280936, −11.69651743178879731853867439517, −10.76967854471354118934391485794, −10.3407588131788838382480704638, −9.53862911878222683490337755776, −8.88769331617858525051020679820, −8.14241543440510636323143252664, −7.42872611267424367569509591241, −5.75602572451604592373180754850, −5.29426837996014694639131052885, −4.032749558538244790801428046398, −3.724044042600749792431449973094, −2.791566188343606937503870083196, −1.929643189246907891036311834330, −0.67893772654752429029879274436, 1.02896249579415345569457984842, 1.617323256646198477237045433157, 3.16430487983466454006348264527, 3.7677747394239982806649529576, 5.13986177447371081873592771101, 5.759636409541933141543492618709, 6.628827602965290769204429115092, 7.27353476202336861921835075532, 7.95539195437180974482128295351, 8.68494004700253623108842040101, 9.34231938989450636154908291871, 10.18480971861492032129241227238, 11.35223924067285552760192337342, 12.109235462154809011216001748558, 13.15994729130877453237264194490, 13.512064478880402205474128874002, 14.3430930731698378155268466155, 14.851438900142630966603396848771, 15.87182279402435560167038972159, 16.38740258649444570087950375854, 17.3785289757459785614130275527, 17.94187720802083177529146988341, 18.57838703622592717804275860213, 19.15654467487557128846587977829, 19.91245538326126205213415929095

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