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Properties

Label 1-775-775.146-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.979 - 0.200i$
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  + (0.309 + 0.951i)2-s + (0.104 − 0.994i)3-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)6-s + (0.669 − 0.743i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.669 − 0.743i)11-s + (0.5 + 0.866i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (0.104 − 0.994i)17-s + (−0.104 − 0.994i)18-s + (0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.104 − 0.994i)3-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)6-s + (0.669 − 0.743i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.669 − 0.743i)11-s + (0.5 + 0.866i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (0.104 − 0.994i)17-s + (−0.104 − 0.994i)18-s + (0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.979 - 0.200i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.979 - 0.200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07582164066 - 0.7477924991i\)
\(L(\frac12)\) \(\approx\) \(0.07582164066 - 0.7477924991i\)
\(L(1)\) \(\approx\) \(0.9825003624 - 0.1014959957i\)
\(L(1)\) \(\approx\) \(0.9825003624 - 0.1014959957i\)

Euler product

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

−22.24840087190712008491983816142, −21.55041756733706082720161470611, −21.063293399197048897757462850809, −20.37555942265325495881490369245, −19.46278701473158353464362732756, −18.7444627877944181377251348212, −17.65569747461156005180428179401, −17.13124560818173075423226421347, −15.68067644022990735514195412162, −15.120060769241350223780547976652, −14.45405034608958459267842124879, −13.56511855674502735238046610644, −12.47521436639417241871132512046, −11.69476054791990911967605265231, −11.0452541788276689313073051095, −10.09617613517276887985213564963, −9.43822964261608605919869812353, −8.68960907238776885022515457587, −7.64353693246350470131527932452, −5.974032970457485885327426275224, −5.02039539134041366622846455387, −4.640184245556650912085817240425, −3.46584696080087196420702606363, −2.52134469087611780035299358425, −1.65954552357161776512436581903, 0.16202907748552373376246118397, 1.059762206290961858539685029564, 2.71118262683462776939282781500, 3.546035939291587386287868180253, 5.14377797144956413904760746254, 5.38748286028688101148394145928, 6.819697506041305305713616633014, 7.34197281865056050673495334646, 8.06912764256738377951132323396, 8.78236120704875356808504042290, 10.05871636345009196223008384601, 11.217796165837240774778814977554, 12.14874094308817680434227562065, 13.02739282582973215210178606165, 13.70205033338545366502524983401, 14.31619836763901694574720927677, 15.073921470614202253907883618815, 16.28149837795614188677953011038, 16.85985611311036973613369886990, 17.85609335669362505404882816912, 18.22461574643936145461861521512, 19.13156582390820241879272165789, 20.29301631007551771052275468166, 20.88007174508311120305025059962, 22.09784644880050509806818479048

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