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Properties

Label 1-1931-1931.1052-r0-0-0
Degree $1$
Conductor $1931$
Sign $0.223 - 0.974i$
Analytic cond. $8.96752$
Root an. cond. $8.96752$
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  + (0.975 + 0.219i)2-s + (0.799 − 0.600i)3-s + (0.903 + 0.428i)4-s + (−0.902 − 0.431i)5-s + (0.911 − 0.410i)6-s + (0.889 + 0.457i)7-s + (0.787 + 0.616i)8-s + (0.277 − 0.960i)9-s + (−0.785 − 0.618i)10-s + (−0.217 − 0.975i)11-s + (0.979 − 0.200i)12-s + (−0.834 − 0.550i)13-s + (0.766 + 0.641i)14-s + (−0.980 + 0.197i)15-s + (0.632 + 0.774i)16-s + (−0.988 + 0.152i)17-s + ⋯
L(s)  = 1  + (0.975 + 0.219i)2-s + (0.799 − 0.600i)3-s + (0.903 + 0.428i)4-s + (−0.902 − 0.431i)5-s + (0.911 − 0.410i)6-s + (0.889 + 0.457i)7-s + (0.787 + 0.616i)8-s + (0.277 − 0.960i)9-s + (−0.785 − 0.618i)10-s + (−0.217 − 0.975i)11-s + (0.979 − 0.200i)12-s + (−0.834 − 0.550i)13-s + (0.766 + 0.641i)14-s + (−0.980 + 0.197i)15-s + (0.632 + 0.774i)16-s + (−0.988 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1931 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1931 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1931\)
Sign: $0.223 - 0.974i$
Analytic conductor: \(8.96752\)
Root analytic conductor: \(8.96752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1931} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1931,\ (0:\ ),\ 0.223 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.763409242 - 2.202451480i\)
\(L(\frac12)\) \(\approx\) \(2.763409242 - 2.202451480i\)
\(L(1)\) \(\approx\) \(2.121632072 - 0.5799640726i\)
\(L(1)\) \(\approx\) \(2.121632072 - 0.5799640726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1931 \( 1 \)
good2 \( 1 + (0.975 + 0.219i)T \)
3 \( 1 + (0.799 - 0.600i)T \)
5 \( 1 + (-0.902 - 0.431i)T \)
7 \( 1 + (0.889 + 0.457i)T \)
11 \( 1 + (-0.217 - 0.975i)T \)
13 \( 1 + (-0.834 - 0.550i)T \)
17 \( 1 + (-0.988 + 0.152i)T \)
19 \( 1 + (-0.496 - 0.868i)T \)
23 \( 1 + (0.962 - 0.270i)T \)
29 \( 1 + (0.470 - 0.882i)T \)
31 \( 1 + (0.728 - 0.685i)T \)
37 \( 1 + (-0.852 + 0.522i)T \)
41 \( 1 + (-0.703 + 0.711i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (-0.330 - 0.943i)T \)
53 \( 1 + (0.840 + 0.542i)T \)
59 \( 1 + (-0.984 - 0.178i)T \)
61 \( 1 + (0.290 - 0.956i)T \)
67 \( 1 + (0.637 + 0.770i)T \)
71 \( 1 + (-0.872 + 0.489i)T \)
73 \( 1 + (0.995 - 0.0975i)T \)
79 \( 1 + (-0.620 - 0.784i)T \)
83 \( 1 + (0.271 - 0.962i)T \)
89 \( 1 + (0.934 - 0.356i)T \)
97 \( 1 + (0.570 + 0.821i)T \)
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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

−20.2323583408076601781472426240, −19.67258432860319560988386871812, −19.19842116820460952988557476014, −18.12678372597215739717852135835, −17.05295736923962498863616543226, −16.22719700186936227299235417997, −15.39556472644947512269374173945, −14.99857388982971378958183513936, −14.36546707096614959409874971328, −13.86553669939512568256588211682, −12.79068215535617708650187274608, −12.10624015982860995196489864793, −11.24818443751469528888059373587, −10.60865595791337838124680282238, −10.04459442114539794786685388348, −8.877719606158376595448015817703, −7.96229859460130970401237069142, −7.215662712657444036294025484542, −6.76981283361881483971174985997, −5.06666848665941576033042016566, −4.66292226094837000501268475459, −4.06860210425839599773701824303, −3.20033616116315899345270974100, −2.33196248156261123596356963719, −1.57530078396626248765636131114, 0.731182638394547313247167029043, 2.0522420712525541149015874479, 2.73432954266819834548140902220, 3.52956322469591278292087322817, 4.571022209744761179918735464280, 5.0287681809769594585225379539, 6.19926926235307219256008188643, 7.02821937219884346061137115348, 7.7723496337749027134099701670, 8.46900486830512707699151434545, 8.811513162271209858716691332113, 10.40153173341590571217164046620, 11.43493924868561174946422963174, 11.76882532255691343661875318752, 12.66036296344090142995621503623, 13.29102049854200581465758328875, 13.87037181930217174993841808218, 14.96862614154496934561500279519, 15.18861523653195651177215504202, 15.762857137460008212337765090, 17.02372336583633283525300134666, 17.46545448332453543716863416119, 18.73537092655165454493383434316, 19.247394117945897121055471361577, 20.08472133641572977341270215148

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