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Properties

Label 1-4033-4033.1965-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.0853 - 0.996i$
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 + (−0.993 + 0.116i)3-s + 4-s + (0.686 − 0.727i)5-s + (0.993 − 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.993 + 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (−0.597 + 0.802i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.993 + 0.116i)3-s + 4-s + (0.686 − 0.727i)5-s + (0.993 − 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.993 + 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (−0.597 + 0.802i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0853 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0853 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0853 - 0.996i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.0853 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6447552941 - 0.7023551544i\)
\(L(\frac12)\) \(\approx\) \(0.6447552941 - 0.7023551544i\)
\(L(1)\) \(\approx\) \(0.6632837893 - 0.1645069851i\)
\(L(1)\) \(\approx\) \(0.6632837893 - 0.1645069851i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (0.686 - 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (0.686 - 0.727i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.973 - 0.230i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.993 - 0.116i)T \)
53 \( 1 + (0.396 - 0.918i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (-0.893 + 0.448i)T \)
67 \( 1 + (-0.993 - 0.116i)T \)
71 \( 1 + T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (-0.597 + 0.802i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (0.286 + 0.957i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−18.44838122969047349898641556808, −18.01173505787499843383858923347, −17.458103715389573632194258900989, −16.85834769701908060637292136078, −16.105639677878257376921704402569, −15.46372135008931753393096413527, −14.73508348821299181192856774033, −13.85878373951446339299838419434, −13.18964243922912386532571859633, −12.14089224515705461351951937173, −11.340121868431780495470916857450, −11.20521855624626352561689043976, −10.42668462423129441457878638018, −9.5709488276282195604582988416, −9.37023927796073841114235564487, −7.82558756698802734847264462050, −7.53824443663095969708053926778, −6.894570577886592640460243111520, −6.023509993707631686776125696505, −5.44715972023637473088552239152, −4.66063639274330354968310105301, −3.459069692201219622262601662179, −2.3953814237513392751512283791, −1.66641032107972039053181912593, −1.06429430478032474214257515696, 0.48886554332486067059672860874, 1.29283176770117969629235622375, 1.842283266770104891783150494958, 2.9974989393827371929827478313, 4.077394464989298815743287798584, 5.240121627590849321858528885160, 5.638730902000529015532556217632, 6.08612599890367689105028940555, 7.18393540531386400126542942827, 8.07366761552890415976841145276, 8.44213342591296999661338412507, 9.29746063687605643660882340989, 10.15589600663302519570446929444, 10.68235255228187023366645231786, 11.11566408049524872328691064026, 12.11488962764556623588554288867, 12.479168529608333982122234765, 13.362173360520397098881309483002, 14.281141081106930984818250891342, 15.21340154152239944280500203509, 15.8750983404001431809119045524, 16.48077758353043592106753848880, 16.962802646891278657223185414794, 17.68764350938508364166921743641, 18.22134557479008574081569394299

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