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Properties

Label 1-5077-5077.32-r1-0-0
Degree $1$
Conductor $5077$
Sign $0.839 + 0.542i$
Analytic cond. $545.599$
Root an. cond. $545.599$
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  + (0.999 + 0.00618i)2-s + (−0.289 + 0.957i)3-s + (0.999 + 0.0123i)4-s + (−0.264 + 0.964i)5-s + (−0.295 + 0.955i)6-s + (−0.595 + 0.803i)7-s + (0.999 + 0.0185i)8-s + (−0.832 − 0.553i)9-s + (−0.270 + 0.962i)10-s + (−0.435 + 0.900i)11-s + (−0.301 + 0.953i)12-s + (0.681 + 0.731i)13-s + (−0.600 + 0.799i)14-s + (−0.846 − 0.531i)15-s + (0.999 + 0.0247i)16-s + (−0.629 − 0.777i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.00618i)2-s + (−0.289 + 0.957i)3-s + (0.999 + 0.0123i)4-s + (−0.264 + 0.964i)5-s + (−0.295 + 0.955i)6-s + (−0.595 + 0.803i)7-s + (0.999 + 0.0185i)8-s + (−0.832 − 0.553i)9-s + (−0.270 + 0.962i)10-s + (−0.435 + 0.900i)11-s + (−0.301 + 0.953i)12-s + (0.681 + 0.731i)13-s + (−0.600 + 0.799i)14-s + (−0.846 − 0.531i)15-s + (0.999 + 0.0247i)16-s + (−0.629 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.839 + 0.542i$
Analytic conductor: \(545.599\)
Root analytic conductor: \(545.599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (1:\ ),\ 0.839 + 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.230056160 + 0.6578197483i\)
\(L(\frac12)\) \(\approx\) \(2.230056160 + 0.6578197483i\)
\(L(1)\) \(\approx\) \(1.255381441 + 0.7425830885i\)
\(L(1)\) \(\approx\) \(1.255381441 + 0.7425830885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (0.999 + 0.00618i)T \)
3 \( 1 + (-0.289 + 0.957i)T \)
5 \( 1 + (-0.264 + 0.964i)T \)
7 \( 1 + (-0.595 + 0.803i)T \)
11 \( 1 + (-0.435 + 0.900i)T \)
13 \( 1 + (0.681 + 0.731i)T \)
17 \( 1 + (-0.629 - 0.777i)T \)
19 \( 1 + (0.517 - 0.855i)T \)
23 \( 1 + (-0.987 - 0.159i)T \)
29 \( 1 + (-0.919 - 0.393i)T \)
31 \( 1 + (0.895 - 0.445i)T \)
37 \( 1 + (-0.0544 - 0.998i)T \)
41 \( 1 + (-0.885 + 0.465i)T \)
43 \( 1 + (-0.323 + 0.946i)T \)
47 \( 1 + (-0.778 - 0.627i)T \)
53 \( 1 + (0.992 + 0.125i)T \)
59 \( 1 + (-0.999 + 0.0420i)T \)
61 \( 1 + (0.965 - 0.261i)T \)
67 \( 1 + (0.00495 - 0.999i)T \)
71 \( 1 + (0.955 + 0.295i)T \)
73 \( 1 + (0.986 - 0.166i)T \)
79 \( 1 + (0.626 + 0.779i)T \)
83 \( 1 + (0.788 - 0.614i)T \)
89 \( 1 + (-0.679 + 0.734i)T \)
97 \( 1 + (0.996 - 0.0890i)T \)
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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

−17.650396201424601653804689363422, −16.9050095381650852317146850801, −16.46030765437513422579648960550, −15.87322418363401038171373235964, −15.21042977875063331168839817800, −14.033405209913298375121175324, −13.59781269195933815185017862535, −13.18955452463883406326742652364, −12.60088799729260753312084067664, −12.00569856178919155424768076705, −11.29560078765428590576569056160, −10.61385420702989511912425686116, −9.96723309732867484860162565912, −8.532155635133501881446279140931, −8.16213609212580843589998222856, −7.48782121413808408359327048933, −6.62406626621541937936670782795, −5.97001456687031364654310167616, −5.51038276618696394913672788655, −4.68922286517490483769452483383, −3.57750079919829830446426290649, −3.423897637885232014187381195596, −2.13564827993758012012404486215, −1.32842133605228885196463630183, −0.69699504403838655173694820942, 0.28397917714699808856646783392, 2.09201384277768849005654581291, 2.52066777147772970111668546432, 3.36084757415149883242833837511, 3.93806139764137356484230198741, 4.72435509914557395847580734045, 5.338328178998464108325056167206, 6.296060194045573358658323851647, 6.55279606454057698972703017548, 7.42908833399859148357418176529, 8.35803463798574623399443616916, 9.47581459556738785159271229145, 9.824896175423144394781037814189, 10.69663051398770881361382365327, 11.457188352075804716151311787848, 11.65201860851903944452161328345, 12.46521379931312801891407011917, 13.43720197596628339439751015902, 13.92787880436248909561950190415, 14.83218067362815218715891541077, 15.26829379668712317718970698692, 15.762399642778332273030214584823, 16.163197992468146049296950094752, 17.01911487476380025636893108653, 18.14799830262173318242070303731

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