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Properties

Label 1-4033-4033.5-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.368 + 0.929i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  − i·2-s + (0.286 + 0.957i)3-s − 4-s + (0.448 + 0.893i)5-s + (0.957 − 0.286i)6-s + (−0.835 − 0.549i)7-s + i·8-s + (−0.835 + 0.549i)9-s + (0.893 − 0.448i)10-s + (−0.893 − 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.448 + 0.893i)13-s + (−0.549 + 0.835i)14-s + (−0.727 + 0.686i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (0.286 + 0.957i)3-s − 4-s + (0.448 + 0.893i)5-s + (0.957 − 0.286i)6-s + (−0.835 − 0.549i)7-s + i·8-s + (−0.835 + 0.549i)9-s + (0.893 − 0.448i)10-s + (−0.893 − 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.448 + 0.893i)13-s + (−0.549 + 0.835i)14-s + (−0.727 + 0.686i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.368 + 0.929i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.428837878 + 0.9707201182i\)
\(L(\frac12)\) \(\approx\) \(1.428837878 + 0.9707201182i\)
\(L(1)\) \(\approx\) \(0.9853673681 + 0.03016885842i\)
\(L(1)\) \(\approx\) \(0.9853673681 + 0.03016885842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (0.448 + 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (-0.893 - 0.448i)T \)
13 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.549 + 0.835i)T \)
31 \( 1 + (-0.802 - 0.597i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (0.727 - 0.686i)T \)
61 \( 1 + (-0.918 - 0.396i)T \)
67 \( 1 + (0.286 - 0.957i)T \)
71 \( 1 + T \)
73 \( 1 + (0.835 - 0.549i)T \)
79 \( 1 + (0.727 - 0.686i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.998 + 0.0581i)T \)
97 \( 1 + (-0.802 + 0.597i)T \)
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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

−18.197644399283377642040588499480, −17.50321275271553340316762067238, −16.84561687133826799520860900617, −16.13994569762699864944668049296, −15.381958963293567955546802326225, −15.00403192976680607640862966528, −13.83746479083039909029832833047, −13.4011621865325036035502936463, −12.78672202130694146722954877156, −12.56225321978340925777090397859, −11.56259406896204447324224805656, −10.18301614302073885949442502987, −9.64578277154559468706202095161, −8.88614468301939917863470851347, −8.36649839477934255574813934758, −7.61103334034142185589043821154, −7.070851912705298162503455242933, −6.0813098927908197456972779606, −5.47853727677346004717733115547, −5.254350877229853864030425483959, −3.80889715843417021365610048900, −3.09047320559329897494539149948, −2.10816121439839899341584081894, −1.029433594567479211453163739487, −0.38154436026765438902012639128, 0.669595600590436342061295637023, 1.911857954459251331738790415040, 2.74364902382921215642617425306, 3.37702039574525565919793913352, 3.731418851784030285542550491253, 4.77215161006818682892677282176, 5.56216583325181245674099883641, 6.23896601574538577451734972328, 7.40300604738715089097850574008, 8.10251620471289315113951512628, 9.20238229582177031352794813431, 9.55051050304699163909659156449, 10.19288730845898794551047218019, 10.87257998726100658687494925923, 11.119396128270198192950497110413, 12.18506791034863615492133339506, 13.076228224566953234893003814472, 13.70264544361920014865712844306, 14.27667905766171491730172927269, 14.68876735610876192304478809925, 15.80836695408209312712471199403, 16.538618925112615852047475467732, 16.84630984460563177123773984637, 18.09997291695290271357003648754, 18.586369881702596357002774676235

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