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Properties

Label 1-775-775.241-r1-0-0
Degree $1$
Conductor $775$
Sign $0.380 + 0.924i$
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  + (−0.809 − 0.587i)2-s + (0.5 − 0.866i)3-s + (0.309 + 0.951i)4-s + (−0.913 + 0.406i)6-s + (0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.978 + 0.207i)12-s + (−0.913 − 0.406i)13-s + (−0.104 − 0.994i)14-s + (−0.809 + 0.587i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.5 − 0.866i)3-s + (0.309 + 0.951i)4-s + (−0.913 + 0.406i)6-s + (0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.978 + 0.207i)12-s + (−0.913 − 0.406i)13-s + (−0.104 − 0.994i)14-s + (−0.809 + 0.587i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.380 + 0.924i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5726615243 + 0.3835131668i\)
\(L(\frac12)\) \(\approx\) \(0.5726615243 + 0.3835131668i\)
\(L(1)\) \(\approx\) \(0.7427281660 - 0.2485337781i\)
\(L(1)\) \(\approx\) \(0.7427281660 - 0.2485337781i\)

Euler product

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

−21.83299890732469184467005173534, −21.15588165097471936528401373523, −20.19616340582784733666956968862, −19.50339671630581997147470541150, −19.050459881274887164155778078410, −17.61937900402866365951562884762, −17.0111666705107947617253103313, −16.60029049681799421419798816844, −15.29106305410859614129034080402, −15.05809385646011995497104795092, −13.994302922469699313895883146, −13.51292987294405333382997797312, −11.66206684206011402786789744132, −10.92205103488442671455354417147, −10.35108345540716838757733355310, −9.24521186992269575555278020262, −8.77297160280932743205337316768, −7.82885326006460285261313858154, −7.02747845395993068225130211280, −5.89673816834886862563055972627, −4.80537098882064877072517369970, −4.12087383945776683163427123743, −2.695292460732027005438801742353, −1.56745179048193430078037526831, −0.191594277471242401843780463203, 1.15350240791588085162755844799, 2.1412096185104157330398215344, 2.646226202829531257587206212592, 3.966340611033558153974549688367, 5.20851645291770784776839637343, 6.664179775832835506304362856788, 7.318029768899131284181113349111, 8.16072423026652544475946937680, 8.965374208967753073421788687377, 9.57182255344703489530593100284, 10.78640998320343618019670659508, 11.68857129125022291058077520568, 12.49010955236551026488098328033, 12.79135770885040528603361656504, 14.25858693420958245182914422348, 14.82852020408015121232831442768, 15.8475591881036348595694692526, 17.17986233538899697127864736583, 17.59686916490067046984214924646, 18.402939716174755551750826172196, 19.00298005750053149097452714113, 19.8734457517415950897941117042, 20.467348736210957623140949309754, 21.17767140481399679490136647585, 22.24313315107689700362898752589

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