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Properties

Label 1-775-775.112-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.926 + 0.376i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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 + (−0.207 + 0.978i)3-s + (−0.309 + 0.951i)4-s + (0.913 − 0.406i)6-s + (0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.104 + 0.994i)11-s + (−0.866 − 0.5i)12-s + (0.994 + 0.104i)13-s + (−0.669 − 0.743i)14-s + (−0.809 − 0.587i)16-s + (0.207 − 0.978i)17-s + (0.207 + 0.978i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.207 + 0.978i)3-s + (−0.309 + 0.951i)4-s + (0.913 − 0.406i)6-s + (0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.104 + 0.994i)11-s + (−0.866 − 0.5i)12-s + (0.994 + 0.104i)13-s + (−0.669 − 0.743i)14-s + (−0.809 − 0.587i)16-s + (0.207 − 0.978i)17-s + (0.207 + 0.978i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.926 + 0.376i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.926 + 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07811284715 + 0.3991039208i\)
\(L(\frac12)\) \(\approx\) \(0.07811284715 + 0.3991039208i\)
\(L(1)\) \(\approx\) \(0.6989428019 + 0.04352445877i\)
\(L(1)\) \(\approx\) \(0.6989428019 + 0.04352445877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (0.994 - 0.104i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.743 + 0.669i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.207 + 0.978i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.994 + 0.104i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−21.88549414237387737075984952367, −20.9474813162116863196715468759, −19.87197057419075224354348107223, −19.03947602487850489026851073795, −18.50448085036614778498537853469, −17.75728746404570796137744594757, −17.119906720212714383946499846387, −16.310322351623544588317701411751, −15.368333805114412845369683416296, −14.32727679569846473468660899177, −13.88037336268336325097028526914, −12.952467947900140606663350853599, −11.77661848844008854754565452018, −10.985269090719055501735843934637, −10.271775354229894614294620241, −8.69617308885009999632578298586, −8.26676520656453224579672513219, −7.73432095356123380680497244731, −6.37253878099004892184339996760, −6.006483704246095919515538220809, −5.06389612506761352873464609289, −3.69256736625848611141736938918, −1.95939943089656718262464382589, −1.31085750140932306655013485219, −0.12288244452021227234839308497, 1.24827673635407823292677470065, 2.37477972714656597431504335509, 3.45915039405370813244127865717, 4.53097569819431522803264500535, 4.920095185661454020772936029375, 6.48588410680930022952464516066, 7.69678216517527775676072663276, 8.57638042279860117746037392879, 9.258090919696811526548903527802, 10.25489658522849116909425678083, 10.7820976510425302167267499804, 11.63348274532329656953153833605, 12.23337274351631067797243397796, 13.526458228057518998433056963626, 14.318775852841277817170860794348, 15.367378379694674642923267789700, 16.13828965879939239471907912152, 16.9996196401457235626864980915, 17.89232159381443474075105272999, 18.15250417890824063432623259290, 19.505686134776796869060760336067, 20.39591922487914606668292939782, 20.77328026146375075167709597316, 21.442085532144099479381120485358, 22.30966477223028285498875410259

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