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Properties

Label 1-4033-4033.1297-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.476 + 0.879i$
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  + (−0.5 + 0.866i)2-s + (−0.396 − 0.918i)3-s + (−0.5 − 0.866i)4-s + (0.957 + 0.286i)5-s + (0.993 + 0.116i)6-s + (−0.686 − 0.727i)7-s + 8-s + (−0.686 + 0.727i)9-s + (−0.727 + 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.597 + 0.802i)12-s + (−0.286 + 0.957i)13-s + (0.973 − 0.230i)14-s + (−0.116 − 0.993i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.396 − 0.918i)3-s + (−0.5 − 0.866i)4-s + (0.957 + 0.286i)5-s + (0.993 + 0.116i)6-s + (−0.686 − 0.727i)7-s + 8-s + (−0.686 + 0.727i)9-s + (−0.727 + 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.597 + 0.802i)12-s + (−0.286 + 0.957i)13-s + (0.973 − 0.230i)14-s + (−0.116 − 0.993i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.476 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5246140196 + 0.8804888448i\)
\(L(\frac12)\) \(\approx\) \(0.5246140196 + 0.8804888448i\)
\(L(1)\) \(\approx\) \(0.7519490295 + 0.2488865081i\)
\(L(1)\) \(\approx\) \(0.7519490295 + 0.2488865081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.396 - 0.918i)T \)
5 \( 1 + (0.957 + 0.286i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (0.727 + 0.686i)T \)
13 \( 1 + (-0.286 + 0.957i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (0.998 + 0.0581i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (-0.116 + 0.993i)T \)
59 \( 1 + (-0.993 + 0.116i)T \)
61 \( 1 + (-0.998 + 0.0581i)T \)
67 \( 1 + (0.802 - 0.597i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (0.396 - 0.918i)T \)
83 \( 1 + (0.0581 + 0.998i)T \)
89 \( 1 + (0.727 - 0.686i)T \)
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

−18.14490662683096883268867690993, −17.66264512568812276900663048979, −16.92086269586709423934760644531, −16.32314669516646957957384157285, −15.89800838349428358891665976484, −14.79827333927995174799362149427, −14.04975935686775692193831292029, −13.37928049805140605648621868946, −12.43564338399995195164078878988, −12.1126504517060037040972245622, −11.23427824322588659506382197373, −10.55494006164738460441119949924, −9.85822987729981321489445829789, −9.40135952702518128985360243717, −8.91109332967981202060212098391, −8.19767791016457364206168698796, −6.918327313698545037527705527256, −6.103798563176520759326679031315, −5.31459037971985025568749783301, −4.83339677392060824021056948104, −3.70982969162392299521946531523, −2.919531743233783577162593786054, −2.59468518014332595204242849050, −1.17574699545133233545576766334, −0.41282298394635575611150770485, 1.2386114602944736327995026207, 1.478226477016272015540009115253, 2.544126544663189741210994659941, 3.83048656632654721035211365013, 4.69601963640666579034749421999, 5.73141627146247105393789354456, 6.20732569674483664066116910256, 6.68889206003681599783108764143, 7.46115409765011405424026464658, 7.8634003622779498929287044118, 9.13497641128436743136087218367, 9.59563526258964770537963270064, 10.20004623381682494849229810522, 10.96071394476124138545454256168, 11.93445541778551117831706629060, 12.64837323711635451887962002841, 13.54296274177661576170321114106, 13.92647349527248735028972853107, 14.40404103988105629719603950161, 15.32032827623673768198602038011, 16.30702898795783291872526148871, 16.92269007281914723402998926658, 17.297420253050543052950293714759, 17.73433595358595896109365073028, 18.70521124231970928913603517305

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