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Properties

Label 1-4033-4033.1112-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.984 - 0.175i$
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  + (0.866 − 0.5i)2-s + (0.686 + 0.727i)3-s + (0.5 − 0.866i)4-s + (0.549 + 0.835i)5-s + (0.957 + 0.286i)6-s + (−0.0581 − 0.998i)7-s − i·8-s + (−0.0581 + 0.998i)9-s + (0.893 + 0.448i)10-s + (−0.893 + 0.448i)11-s + (0.973 − 0.230i)12-s + (0.549 + 0.835i)13-s + (−0.549 − 0.835i)14-s + (−0.230 + 0.973i)15-s + (−0.5 − 0.866i)16-s − i·17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.686 + 0.727i)3-s + (0.5 − 0.866i)4-s + (0.549 + 0.835i)5-s + (0.957 + 0.286i)6-s + (−0.0581 − 0.998i)7-s − i·8-s + (−0.0581 + 0.998i)9-s + (0.893 + 0.448i)10-s + (−0.893 + 0.448i)11-s + (0.973 − 0.230i)12-s + (0.549 + 0.835i)13-s + (−0.549 − 0.835i)14-s + (−0.230 + 0.973i)15-s + (−0.5 − 0.866i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007648122771 - 0.08629062883i\)
\(L(\frac12)\) \(\approx\) \(0.007648122771 - 0.08629062883i\)
\(L(1)\) \(\approx\) \(1.887584806 + 0.008472049265i\)
\(L(1)\) \(\approx\) \(1.887584806 + 0.008472049265i\)

Euler product

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

−18.39697021988773634941753919162, −18.00043795411853140487317849092, −17.266964917035851536715982888, −16.376535543274979744813485204450, −15.795620046122714383060439609031, −15.0238336229098469505629202181, −14.66912424632272071731686641057, −13.51282574198131909738396807169, −13.27191927783381346638689786634, −12.684183574913129934957822632207, −12.2612915533789997694161739469, −11.275196842120150542238696830855, −10.43580423684833660625438422134, −9.142739594262747851792696130439, −8.74247911185636787799709717433, −8.14776371365017748191163954341, −7.512917968093439516913124951990, −6.44443384932299074719742115940, −5.88517485929684353416892265977, −5.36432985565117814082926583782, −4.54109681685144988292838886416, −3.399455350129871555597461179375, −2.82504599947320342129021882401, −2.122532965741613347192343055846, −1.25384465355614384976094553580, 0.00642982948908420276152877252, 1.60327459279233691324657670267, 2.0252716745624896522971668912, 3.1475477240153176946603085679, 3.44924335830627612693899392810, 4.255869324736051070641616973204, 5.10073652283137558039525106425, 5.65994454842925388570006205239, 6.84685895979752920473653263146, 7.21633611051744829413774373431, 8.131648751782664259616416698266, 9.43551058582836240969309073244, 9.85262239147182458721906119734, 10.30935474465214668923382722595, 11.17647097228550149761137192169, 11.45796304115247545560456928606, 12.78275723559283630139624996752, 13.58119707550579908822008712205, 13.752631873898819019929130383329, 14.42550719008932048944362326431, 15.05521629882616471886106755204, 15.7835711791508792714186743876, 16.349060648515885858539398824995, 17.16608535207913630006171366542, 18.3629149245933677418181508757

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