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Properties

Label 1-4033-4033.2552-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.849 - 0.527i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 − 0.866i)15-s + 16-s − 17-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 − 0.866i)15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.849 - 0.527i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2552, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01514022865 - 0.05309369379i\)
\(L(\frac12)\) \(\approx\) \(0.01514022865 - 0.05309369379i\)
\(L(1)\) \(\approx\) \(0.4802179308 + 0.01372398414i\)
\(L(1)\) \(\approx\) \(0.4802179308 + 0.01372398414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−18.47208505175788884093426533953, −17.976373511049422812687037649311, −17.40800999074286263175187185355, −16.66924248899289161172666110116, −16.11392827998301431270813014628, −15.57052554804196178863022259254, −15.24046188187926262439662076623, −14.019006443051520687766637109744, −13.06146002875690940275738958812, −12.44387428079440948022912011179, −11.724106071569568328698789758469, −11.00000827073966579692961800453, −10.28532344537141751294243292573, −9.83567572797748637339929220679, −8.93417338404490881734152654860, −8.57916325070177905669325509358, −8.045450330271774007786866427538, −6.45351898559243950859279435738, −6.142335126507139067659252639946, −5.53460776599129686045505433542, −4.69851069797297944395422616312, −3.61794285879583040524190907350, −2.76487621257395039685972625589, −2.01400421705343694719794628094, −0.752580984700386401028972212722, 0.0306469582075807605722724871, 1.315971814862789516357355373845, 2.10316712329290199818371286980, 2.53422261701770143110246830175, 3.71745057306170418752791857682, 4.73840517347939495101894392092, 6.02938923156127882777885331271, 6.49189872123750665531983260323, 6.88893035319280458087899537535, 7.57909442349153775202822984460, 8.29730530331928137531206005827, 9.2476179435305721850137951553, 10.0831063291621272388628455418, 10.50709276480642457260888719879, 11.1998682431164921945488663344, 11.73571312685706249557560649176, 12.82930283402660187637860829086, 13.19605191344599764688630403266, 14.13570316730203027071064831141, 14.734662834917724284317407643664, 15.804288194262508279488354848352, 16.28224101358089746074272870670, 17.16715391314885969191688523529, 17.62735241980702937861108599432, 18.06294707211810597265198256192

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