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Properties

Label 1-775-775.188-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.990 + 0.140i$
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 + (−0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.587 + 0.809i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s − i·12-s + (0.587 − 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + (0.951 + 0.309i)18-s + (−0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.587 + 0.809i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s − i·12-s + (0.587 − 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + (0.951 + 0.309i)18-s + (−0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.990 + 0.140i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.990 + 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1264172648 + 1.796796111i\)
\(L(\frac12)\) \(\approx\) \(0.1264172648 + 1.796796111i\)
\(L(1)\) \(\approx\) \(0.8186805530 + 0.7408970756i\)
\(L(1)\) \(\approx\) \(0.8186805530 + 0.7408970756i\)

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 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 - T \)
83 \( 1 + (0.951 + 0.309i)T \)
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

−21.57987482729513691588716623584, −21.14516054944938545228762557783, −20.39854269326876081803443743684, −19.254260704177701071092797183887, −18.70262929523721241070665029654, −17.764972748391331814516221228502, −17.159076158852770035895540603944, −16.06873332019529755190021834428, −15.18214692806466507685795900290, −14.13547708555981897239782124849, −13.39349318426703750305892977343, −12.692275203097800687624305817878, −11.83441203182518451911266812753, −11.041497595011920976368181962227, −10.52626949210041055209827315955, −9.68833935815646271126887944292, −8.32822175151044907065479718850, −7.14238432021740514162400809030, −6.413875037364998339658758638424, −5.199350729407790717708773169598, −4.71018346545278087232099666654, −3.75674111864479329197124069151, −2.32858204047007988197088526705, −1.35080961214568227846714471395, −0.46133272304351415124291507678, 1.02956536843827068202769601225, 2.80835587189642476978328942718, 3.75169711816767034358898884024, 4.99547265748647966260961657593, 5.5099848484828627591618992912, 6.06937849402789079327353320223, 7.286836517730145619390428055047, 8.15922620401241072589622189272, 8.961721640154328009196187152052, 10.27250138166938786786396849078, 11.09252572196000057568572745411, 12.065310029317847560737068835299, 12.59907108446930321925208114764, 13.55953186867600096179711037706, 14.62435944390547191072145421416, 15.34668375125475117574020988521, 16.02535241752210517203168388769, 16.66682115836821044355918729051, 17.63943854809496999904117132500, 18.230613856327549702463759396126, 18.9021032165300415045499539374, 20.795764155091744677848469158400, 21.10075866769041796789193392431, 21.80121635451692137920884923580, 22.74893984864529879897109908360

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