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Properties

Label 1-5077-5077.1069-r0-0-0
Degree $1$
Conductor $5077$
Sign $-0.354 - 0.935i$
Analytic cond. $23.5774$
Root an. cond. $23.5774$
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  + (0.778 − 0.627i)2-s + (0.867 + 0.496i)3-s + (0.212 − 0.977i)4-s + (0.920 − 0.390i)5-s + (0.987 − 0.157i)6-s + (0.729 − 0.683i)7-s + (−0.447 − 0.894i)8-s + (0.506 + 0.862i)9-s + (0.471 − 0.881i)10-s + (−0.967 − 0.252i)11-s + (0.669 − 0.742i)12-s + (−0.995 + 0.0964i)13-s + (0.139 − 0.990i)14-s + (0.992 + 0.118i)15-s + (−0.909 − 0.415i)16-s + (0.709 − 0.704i)17-s + ⋯
L(s)  = 1  + (0.778 − 0.627i)2-s + (0.867 + 0.496i)3-s + (0.212 − 0.977i)4-s + (0.920 − 0.390i)5-s + (0.987 − 0.157i)6-s + (0.729 − 0.683i)7-s + (−0.447 − 0.894i)8-s + (0.506 + 0.862i)9-s + (0.471 − 0.881i)10-s + (−0.967 − 0.252i)11-s + (0.669 − 0.742i)12-s + (−0.995 + 0.0964i)13-s + (0.139 − 0.990i)14-s + (0.992 + 0.118i)15-s + (−0.909 − 0.415i)16-s + (0.709 − 0.704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $-0.354 - 0.935i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ -0.354 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.804087193 - 4.061100999i\)
\(L(\frac12)\) \(\approx\) \(2.804087193 - 4.061100999i\)
\(L(1)\) \(\approx\) \(2.192729656 - 1.252211444i\)
\(L(1)\) \(\approx\) \(2.192729656 - 1.252211444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (0.778 - 0.627i)T \)
3 \( 1 + (0.867 + 0.496i)T \)
5 \( 1 + (0.920 - 0.390i)T \)
7 \( 1 + (0.729 - 0.683i)T \)
11 \( 1 + (-0.967 - 0.252i)T \)
13 \( 1 + (-0.995 + 0.0964i)T \)
17 \( 1 + (0.709 - 0.704i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
23 \( 1 + (-0.751 - 0.659i)T \)
29 \( 1 + (0.639 + 0.768i)T \)
31 \( 1 + (-0.0235 + 0.999i)T \)
37 \( 1 + (-0.950 - 0.309i)T \)
41 \( 1 + (0.548 - 0.836i)T \)
43 \( 1 + (0.620 - 0.784i)T \)
47 \( 1 + (0.979 - 0.201i)T \)
53 \( 1 + (0.999 - 0.0148i)T \)
59 \( 1 + (-0.504 + 0.863i)T \)
61 \( 1 + (0.183 + 0.983i)T \)
67 \( 1 + (-0.389 + 0.921i)T \)
71 \( 1 + (0.987 - 0.157i)T \)
73 \( 1 + (0.860 - 0.509i)T \)
79 \( 1 + (-0.767 - 0.640i)T \)
83 \( 1 + (-0.947 - 0.318i)T \)
89 \( 1 + (0.188 - 0.982i)T \)
97 \( 1 + (-0.610 + 0.791i)T \)
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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

−18.2864944589050074324201372607, −17.47107791977598399515564477112, −17.11853181862791924760317361847, −15.91488008249987286701712732190, −15.30389918387120965746606327845, −14.8401496800321970380342817022, −14.095627611709960905115997488861, −13.89354255517584581503680114966, −12.99514524045367723607821983020, −12.40972387811855423886037839788, −11.92184058057588501558209034632, −10.91935614145393168696726712148, −9.809936490300865720816892447799, −9.52597990277314378946478141453, −8.33943849957267097175598607194, −7.87257441924286974327575383270, −7.45439016726201360924856633787, −6.50377589634534942573891848629, −5.730026089809638480282134937895, −5.331156145777455449783558182537, −4.40172561191241619863387639359, −3.444383238854364916927385653603, −2.58435901113936458823196306878, −2.30901888379817131441200152615, −1.424985250698631115469203821573, 0.77184377181465974660583322631, 1.637557199763619416249649642499, 2.50524759993254426980668902431, 2.837596281336950219663445255037, 3.883612169112511665980246842450, 4.63734591408940922016355150210, 5.24172878911288255742468163502, 5.542721030733030130965813851314, 7.05383477831054215156980366164, 7.397053406723915690154923133268, 8.523736128499996433479511302700, 9.132454472761495111316736209338, 9.97190722987351943173458665453, 10.398985036914766536225256055209, 10.79893944240170416433339111736, 12.06122212653983569379322393865, 12.433721757884447937709496651884, 13.552200977543636860311067949914, 13.73231230612401136487578554134, 14.319333301372642432332409248325, 14.77309946533018086131830434368, 15.92128495394346879476697928191, 16.11994752967597176248753446751, 17.13829032188936543234930142367, 18.012425545420945570317542289846

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