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Properties

Label 1-4033-4033.705-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.316 + 0.948i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.939 − 0.342i)2-s + (−0.973 − 0.230i)3-s + (0.766 + 0.642i)4-s + (−0.448 − 0.893i)5-s + (0.835 + 0.549i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (0.116 + 0.993i)10-s + (−0.116 + 0.993i)11-s + (−0.597 − 0.802i)12-s + (−0.0581 + 0.998i)13-s + (0.597 + 0.802i)14-s + (0.230 + 0.973i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.973 − 0.230i)3-s + (0.766 + 0.642i)4-s + (−0.448 − 0.893i)5-s + (0.835 + 0.549i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (0.116 + 0.993i)10-s + (−0.116 + 0.993i)11-s + (−0.597 − 0.802i)12-s + (−0.0581 + 0.998i)13-s + (0.597 + 0.802i)14-s + (0.230 + 0.973i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (705, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2096878941 + 0.1510070374i\)
\(L(\frac12)\) \(\approx\) \(0.2096878941 + 0.1510070374i\)
\(L(1)\) \(\approx\) \(0.4046676266 - 0.09929647673i\)
\(L(1)\) \(\approx\) \(0.4046676266 - 0.09929647673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (-0.448 - 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (-0.116 + 0.993i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.116 + 0.993i)T \)
31 \( 1 + (0.957 + 0.286i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.998 + 0.0581i)T \)
53 \( 1 + (0.802 - 0.597i)T \)
59 \( 1 + (0.973 - 0.230i)T \)
61 \( 1 + (0.116 + 0.993i)T \)
67 \( 1 + (0.918 - 0.396i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (-0.993 - 0.116i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (0.957 - 0.286i)T \)
97 \( 1 + (0.230 - 0.973i)T \)
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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

−18.46391858448931942805679224536, −17.58935790160706026870555872665, −17.04752143822675594366757903818, −16.24579709666593595315252192297, −15.74506675605097736928487313489, −15.33691337965710232245576892532, −14.56658124065581812773164654669, −13.6158330091700022558595654965, −12.59011405119708239609179400391, −11.81217083621853275866034030370, −11.43955379877749471688874619266, −10.57257765446467340431173971340, −10.0216466651358866804926649210, −9.66871938013563955886238774701, −8.424626646851802778695137450847, −7.89135146046150681519527158544, −7.071877930498295898158032365824, −6.338761100525955520464842655373, −5.784891565643790866946836672393, −5.37496156197272837154779624108, −3.86478420424569792719791938827, −3.1845581810587465421223031753, −2.39271422183264761739420403031, −1.046715217542444524312007088685, −0.18084758154150625426126430817, 0.80374967716804220231465446425, 1.50673618265335641730628682618, 2.43180971963671604866833240626, 3.647550561571696767104620432117, 4.37062331211466091592512803326, 5.02614249982985633048276299734, 6.23310684783056465794168234857, 6.84676402853349179112659696319, 7.30481977777108295121394498874, 8.20512373974569308752430306670, 8.95470467272116920786156887530, 9.81313340841205203925534560418, 10.19219925803213575627181165761, 11.010075856804495703854796903937, 11.79639673883111478613835899729, 12.33574931148552402048225026476, 12.77038428360600351961675849100, 13.42159538175966080085152440596, 14.711181095807872244389874214611, 15.69461159960973844089298444161, 16.17973158455996017696649998546, 16.72672594574459275856993474156, 17.14172218016981734739612978164, 17.82706154013565083577661499368, 18.626562664201557527151298794821

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