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Properties

Label 1-71e2-5041.128-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.757 - 0.652i$
Analytic cond. $23.4103$
Root an. cond. $23.4103$
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  + (0.925 − 0.379i)2-s + (0.998 + 0.0530i)3-s + (0.711 − 0.702i)4-s + (0.853 − 0.521i)5-s + (0.943 − 0.329i)6-s + (−0.416 + 0.909i)7-s + (0.391 − 0.920i)8-s + (0.994 + 0.105i)9-s + (0.591 − 0.806i)10-s + (−0.809 − 0.587i)11-s + (0.748 − 0.663i)12-s + (0.925 + 0.379i)13-s + (−0.0398 + 0.999i)14-s + (0.879 − 0.475i)15-s + (0.0132 − 0.999i)16-s + (0.375 + 0.926i)17-s + ⋯
L(s)  = 1  + (0.925 − 0.379i)2-s + (0.998 + 0.0530i)3-s + (0.711 − 0.702i)4-s + (0.853 − 0.521i)5-s + (0.943 − 0.329i)6-s + (−0.416 + 0.909i)7-s + (0.391 − 0.920i)8-s + (0.994 + 0.105i)9-s + (0.591 − 0.806i)10-s + (−0.809 − 0.587i)11-s + (0.748 − 0.663i)12-s + (0.925 + 0.379i)13-s + (−0.0398 + 0.999i)14-s + (0.879 − 0.475i)15-s + (0.0132 − 0.999i)16-s + (0.375 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.757 - 0.652i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.782658286 - 2.145985365i\)
\(L(\frac12)\) \(\approx\) \(5.782658286 - 2.145985365i\)
\(L(1)\) \(\approx\) \(2.886103573 - 0.7531370216i\)
\(L(1)\) \(\approx\) \(2.886103573 - 0.7531370216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.925 - 0.379i)T \)
3 \( 1 + (0.998 + 0.0530i)T \)
5 \( 1 + (0.853 - 0.521i)T \)
7 \( 1 + (-0.416 + 0.909i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.925 + 0.379i)T \)
17 \( 1 + (0.375 + 0.926i)T \)
19 \( 1 + (-0.819 + 0.573i)T \)
23 \( 1 + (0.996 + 0.0883i)T \)
29 \( 1 + (0.994 - 0.105i)T \)
31 \( 1 + (-0.907 + 0.420i)T \)
37 \( 1 + (0.996 + 0.0883i)T \)
41 \( 1 + (-0.975 + 0.219i)T \)
43 \( 1 + (0.136 - 0.990i)T \)
47 \( 1 + (-0.479 + 0.877i)T \)
53 \( 1 + (-0.680 - 0.733i)T \)
59 \( 1 + (0.803 - 0.594i)T \)
61 \( 1 + (-0.494 - 0.868i)T \)
67 \( 1 + (0.871 - 0.491i)T \)
73 \( 1 + (0.605 - 0.795i)T \)
79 \( 1 + (-0.266 + 0.963i)T \)
83 \( 1 + (0.686 - 0.727i)T \)
89 \( 1 + (-0.180 + 0.983i)T \)
97 \( 1 + (0.903 + 0.428i)T \)
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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

−18.131669679195995538603852846760, −17.40064736170097478523841685077, −16.5851064906479221174274026442, −15.95347574617233048917878072461, −15.233266314682360100394481336558, −14.71847300990786028481179083009, −14.057438183659270980432183917380, −13.37266936907093651688334556099, −13.161152079579851646581330359726, −12.598407639681768965114062375295, −11.31710501405559174778978250892, −10.64911735348417706292086118928, −10.10048018752549807850110918324, −9.302725380952652564498985020922, −8.44074935390750658981221943961, −7.660911429182413565321440389712, −7.00829605045651878338826796699, −6.64330635487396740819679990272, −5.69304010990658604715453980026, −4.83128734694217910634999704171, −4.18418629247956171573352893825, −3.19238834156797479368192905347, −2.88553975946053750833742180759, −2.10553131823408086081061365143, −1.12058887685892151842348782554, 1.104444057503519049699116685838, 1.877060461325518626005490311871, 2.41648837144989748715788867161, 3.263002859693975627215783347430, 3.75215373500525206990740623005, 4.842716233435838965580130579922, 5.36022394283676674950250891599, 6.30786635923533960904762563799, 6.521119308911452305970106731894, 7.94465029501209250840911657770, 8.51712875688421292439297176918, 9.168414546510281642292627444620, 9.888467871963160049448653923611, 10.54005379334500615985814297925, 11.19024446658788063638610192762, 12.45743280145503197750071850570, 12.6555198313418415473685119187, 13.335651569870729008036679230727, 13.77765475592354333222779444071, 14.625944441935628781324045217910, 15.034940693273258465516048739674, 16.00746873473612364420773179044, 16.1589935433086797229766380910, 17.18378586927874400676375471463, 18.42480642996715987017723170766

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