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Properties

Label 1-4024-4024.2299-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.841 - 0.539i$
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  + (0.0187 − 0.999i)3-s + (0.640 + 0.768i)5-s + (−0.998 + 0.0500i)7-s + (−0.999 − 0.0375i)9-s + (0.620 − 0.784i)11-s + (0.253 − 0.967i)13-s + (0.780 − 0.625i)15-s + (−0.910 + 0.412i)17-s + (−0.955 − 0.295i)19-s + (0.0312 + 0.999i)21-s + (0.131 + 0.991i)23-s + (−0.180 + 0.983i)25-s + (−0.0562 + 0.998i)27-s + (−0.704 + 0.709i)29-s + (−0.441 + 0.897i)31-s + ⋯
L(s)  = 1  + (0.0187 − 0.999i)3-s + (0.640 + 0.768i)5-s + (−0.998 + 0.0500i)7-s + (−0.999 − 0.0375i)9-s + (0.620 − 0.784i)11-s + (0.253 − 0.967i)13-s + (0.780 − 0.625i)15-s + (−0.910 + 0.412i)17-s + (−0.955 − 0.295i)19-s + (0.0312 + 0.999i)21-s + (0.131 + 0.991i)23-s + (−0.180 + 0.983i)25-s + (−0.0562 + 0.998i)27-s + (−0.704 + 0.709i)29-s + (−0.441 + 0.897i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.841 - 0.539i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (2299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.841 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344232867 - 0.3940123499i\)
\(L(\frac12)\) \(\approx\) \(1.344232867 - 0.3940123499i\)
\(L(1)\) \(\approx\) \(0.9734899254 - 0.2139773707i\)
\(L(1)\) \(\approx\) \(0.9734899254 - 0.2139773707i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.0187 - 0.999i)T \)
5 \( 1 + (0.640 + 0.768i)T \)
7 \( 1 + (-0.998 + 0.0500i)T \)
11 \( 1 + (0.620 - 0.784i)T \)
13 \( 1 + (0.253 - 0.967i)T \)
17 \( 1 + (-0.910 + 0.412i)T \)
19 \( 1 + (-0.955 - 0.295i)T \)
23 \( 1 + (0.131 + 0.991i)T \)
29 \( 1 + (-0.704 + 0.709i)T \)
31 \( 1 + (-0.441 + 0.897i)T \)
37 \( 1 + (-0.106 - 0.994i)T \)
41 \( 1 + (-0.118 + 0.992i)T \)
43 \( 1 + (0.289 + 0.957i)T \)
47 \( 1 + (0.325 - 0.945i)T \)
53 \( 1 + (0.955 - 0.295i)T \)
59 \( 1 + (0.429 - 0.902i)T \)
61 \( 1 + (0.772 - 0.635i)T \)
67 \( 1 + (0.780 + 0.625i)T \)
71 \( 1 + (0.900 - 0.435i)T \)
73 \( 1 + (-0.998 - 0.0625i)T \)
79 \( 1 + (0.958 - 0.283i)T \)
83 \( 1 + (0.192 + 0.981i)T \)
89 \( 1 + (0.817 - 0.575i)T \)
97 \( 1 + (-0.977 - 0.211i)T \)
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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

−18.62564475603700663092776339724, −17.54571454047944988820575029724, −16.89827037031814490480869465815, −16.68050086370282335108063786352, −15.87116805416992743915549947878, −15.245627232991249322292876124707, −14.51481295685000935764321845206, −13.72370178388616127363081922650, −13.16523941756819878913468509420, −12.33598157334183819314706038698, −11.72628606703671108226136556916, −10.781778456428775212092104506675, −10.09059997832330721006119542772, −9.47496295535915069627999936506, −8.99549266410388596397004980128, −8.518085405853889300169007309751, −7.166248390578430790763591975819, −6.39364029366172006202808297517, −5.88936084918579273805889302206, −4.90125632909165822947999318658, −4.14876033496818506766492024689, −3.89662261937369767114784030667, −2.456076290543557402499658062643, −2.05868835203069200422158692534, −0.59884533511738030122549051682, 0.62864528158615488175622266730, 1.712911913593227726904368898259, 2.43064908070597198774760061765, 3.288543713432982337159794069470, 3.7037619462855228107201838926, 5.34571775538020561175658624639, 5.86825077307281366103212846637, 6.622785937992837288965179401926, 6.86789031780093705890644120224, 7.86070352140062958834596234122, 8.76722977447196896070937446120, 9.2205952429421993316221049868, 10.2175045906086491098030407628, 10.97619807751666979678863198962, 11.377075421686750405032448138640, 12.522492443847549065316947535287, 13.09999173125044933556096258500, 13.38246618964966344446788952628, 14.27340527937356180655524051052, 14.85156061601335242220633900981, 15.63664864996834932530512696759, 16.537454041923045538514994396846, 17.235805865501028078919178196795, 17.82218946850113902665137434055, 18.35972720351961257287790372839

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