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Properties

Label 1-4033-4033.29-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.995 + 0.0927i$
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  + (−0.984 − 0.173i)2-s + (−0.973 − 0.230i)3-s + (0.939 + 0.342i)4-s + (−0.549 + 0.835i)5-s + (0.918 + 0.396i)6-s + (−0.0581 + 0.998i)7-s + (−0.866 − 0.5i)8-s + (0.893 + 0.448i)9-s + (0.686 − 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.835 − 0.549i)12-s + (−0.998 − 0.0581i)13-s + (0.230 − 0.973i)14-s + (0.727 − 0.686i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.973 − 0.230i)3-s + (0.939 + 0.342i)4-s + (−0.549 + 0.835i)5-s + (0.918 + 0.396i)6-s + (−0.0581 + 0.998i)7-s + (−0.866 − 0.5i)8-s + (0.893 + 0.448i)9-s + (0.686 − 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.835 − 0.549i)12-s + (−0.998 − 0.0581i)13-s + (0.230 − 0.973i)14-s + (0.727 − 0.686i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.995 + 0.0927i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.995 + 0.0927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3851630454 + 0.01790786824i\)
\(L(\frac12)\) \(\approx\) \(0.3851630454 + 0.01790786824i\)
\(L(1)\) \(\approx\) \(0.4009181465 + 0.07240746801i\)
\(L(1)\) \(\approx\) \(0.4009181465 + 0.07240746801i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (-0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (-0.998 - 0.0581i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.957 - 0.286i)T \)
31 \( 1 + (-0.549 - 0.835i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (0.0581 - 0.998i)T \)
59 \( 1 + (0.727 - 0.686i)T \)
61 \( 1 + (0.918 + 0.396i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.686 + 0.727i)T \)
79 \( 1 + (-0.549 - 0.835i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (-0.802 + 0.597i)T \)
97 \( 1 + (0.998 + 0.0581i)T \)
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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

−18.07675391309461272152831313788, −17.495773498992846629081125237327, −16.96939793851075268787200282141, −16.39132777066391401147691452286, −15.8106325191585824349963175369, −15.362964260635479867400094442215, −14.380128216232095093739377014654, −13.36027015578715324280705868697, −12.432632818079480181081605619514, −12.09370134460838421027835230784, −11.2636945465432797925962087740, −10.604508746289581269447404276556, −10.02110162532148762243155973182, −9.398624955806634658883490679718, −8.61675781026205996718274117916, −7.61474942622664047789908782540, −7.09566064751540243880206781925, −6.76489077814464488160823339810, −5.27604899180557952748585978242, −5.06159863356590478252807786041, −4.24936441841644157800362158094, −3.130543473933966070178700618445, −2.002115990074967176877153512222, −0.970907355045252925021015460241, −0.422735288489720737211630416716, 0.23873011746683031944194585560, 1.28960958483248718149696436192, 2.372143827995546279076062302632, 2.85482301307974563303823802960, 3.84126384187210856657177632566, 5.08875358637554230414229321131, 5.75851640206813866775703220092, 6.51452735559127543580184932254, 7.08872111309408271602569371631, 7.950847994094061864904552702887, 8.310422057940425018692115784700, 9.6114800271530288894039124495, 9.94069678656866473089692492005, 10.91339686608837992125056563551, 11.39050900534699580787483671756, 11.75429929471648002833465929435, 12.62993247661528782497672260066, 13.169769377318547251238396080389, 14.51691283487611990683500270766, 15.18977844516555828772400902982, 15.877295660212760714141194215395, 16.16629069466260101349586360163, 17.19267152916711815491175784515, 17.78004656953184098688846773747, 18.24885640442127860616273117130

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