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Properties

Label 1-4024-4024.1213-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.398 + 0.917i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.772 + 0.635i)3-s + (0.943 − 0.331i)5-s + (−0.704 + 0.709i)7-s + (0.192 + 0.981i)9-s + (0.0562 + 0.998i)11-s + (0.485 − 0.874i)13-s + (0.939 + 0.343i)15-s + (0.407 + 0.913i)17-s + (−0.0187 + 0.999i)19-s + (−0.994 + 0.0999i)21-s + (−0.106 − 0.994i)23-s + (0.780 − 0.625i)25-s + (−0.474 + 0.880i)27-s + (0.803 − 0.595i)29-s + (0.496 − 0.868i)31-s + ⋯
L(s)  = 1  + (0.772 + 0.635i)3-s + (0.943 − 0.331i)5-s + (−0.704 + 0.709i)7-s + (0.192 + 0.981i)9-s + (0.0562 + 0.998i)11-s + (0.485 − 0.874i)13-s + (0.939 + 0.343i)15-s + (0.407 + 0.913i)17-s + (−0.0187 + 0.999i)19-s + (−0.994 + 0.0999i)21-s + (−0.106 − 0.994i)23-s + (0.780 − 0.625i)25-s + (−0.474 + 0.880i)27-s + (0.803 − 0.595i)29-s + (0.496 − 0.868i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.398 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.520268823 + 1.652994490i\)
\(L(\frac12)\) \(\approx\) \(2.520268823 + 1.652994490i\)
\(L(1)\) \(\approx\) \(1.575502539 + 0.5168131257i\)
\(L(1)\) \(\approx\) \(1.575502539 + 0.5168131257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.772 + 0.635i)T \)
5 \( 1 + (0.943 - 0.331i)T \)
7 \( 1 + (-0.704 + 0.709i)T \)
11 \( 1 + (0.0562 + 0.998i)T \)
13 \( 1 + (0.485 - 0.874i)T \)
17 \( 1 + (0.407 + 0.913i)T \)
19 \( 1 + (-0.0187 + 0.999i)T \)
23 \( 1 + (-0.106 - 0.994i)T \)
29 \( 1 + (0.803 - 0.595i)T \)
31 \( 1 + (0.496 - 0.868i)T \)
37 \( 1 + (0.958 + 0.283i)T \)
41 \( 1 + (0.640 - 0.768i)T \)
43 \( 1 + (0.951 - 0.307i)T \)
47 \( 1 + (0.118 - 0.992i)T \)
53 \( 1 + (-0.0187 - 0.999i)T \)
59 \( 1 + (0.894 - 0.446i)T \)
61 \( 1 + (0.590 + 0.806i)T \)
67 \( 1 + (-0.939 + 0.343i)T \)
71 \( 1 + (-0.686 + 0.726i)T \)
73 \( 1 + (0.980 + 0.198i)T \)
79 \( 1 + (-0.570 - 0.821i)T \)
83 \( 1 + (0.301 + 0.953i)T \)
89 \( 1 + (0.852 + 0.523i)T \)
97 \( 1 + (0.838 - 0.544i)T \)
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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

−18.35917657336992938662341549409, −17.80235186028359891465483254879, −17.11428796210250580755376452409, −16.16513036296953644732627077540, −15.84085982681127170799615662364, −14.56314713498670879465424879912, −14.097381942118111098712710188882, −13.58178720928448735371770561833, −13.24218320467701187483905445648, −12.35467122516469317273411500194, −11.40711026579002772174990817930, −10.78829549203142088748022968685, −9.82626906913804550150066032828, −9.253315041346867165351561666354, −8.841847838495167039133335866869, −7.71221850082831213737850857670, −7.12014788257598770570175458005, −6.39178805295585413150228397258, −6.00644583499166103042468678587, −4.81074589618052691208193946668, −3.82525139559543079177513837284, −2.95087121179361439023160535130, −2.69047270291681873557254750611, −1.38555882660426969735312525309, −0.88973629019389615549007369006, 1.061855267803989150733768522418, 2.32435099155899177479666625311, 2.40196772699610248399327817344, 3.59872954611675894157580199007, 4.233822419326410159405251620739, 5.1872312152205207447505521639, 5.870486043174386916109109508655, 6.43229339148019882614397456819, 7.62397413805936194015610434687, 8.39163239754966749637167460380, 8.838778596658306316824602813205, 9.87088660967656466323887096412, 9.983801903135925608923559345720, 10.62494188952716131171034432229, 11.92635697832645678290263965361, 12.70973896137137438444216279729, 13.033663882984197112259834952261, 13.85097627462552597949593225564, 14.75060985859468417641122311065, 14.99357152232916703085031466175, 15.92191867635168523829138245825, 16.40468883201973637659705779893, 17.22509800642706510332146099062, 17.88346651231537368135899804252, 18.70905370884814851867342438520

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