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Properties

Label 1-4033-4033.524-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.391 + 0.920i$
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  + (0.342 − 0.939i)2-s + (0.286 − 0.957i)3-s + (−0.766 − 0.642i)4-s + (0.998 − 0.0581i)5-s + (−0.802 − 0.597i)6-s + (0.893 + 0.448i)7-s + (−0.866 + 0.5i)8-s + (−0.835 − 0.549i)9-s + (0.286 − 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.835 + 0.549i)12-s + (0.448 − 0.893i)13-s + (0.727 − 0.686i)14-s + (0.230 − 0.973i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.286 − 0.957i)3-s + (−0.766 − 0.642i)4-s + (0.998 − 0.0581i)5-s + (−0.802 − 0.597i)6-s + (0.893 + 0.448i)7-s + (−0.866 + 0.5i)8-s + (−0.835 − 0.549i)9-s + (0.286 − 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.835 + 0.549i)12-s + (0.448 − 0.893i)13-s + (0.727 − 0.686i)14-s + (0.230 − 0.973i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.391 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-2.100427586 - 3.177212271i\)
\(L(\frac12)\) \(\approx\) \(-2.100427586 - 3.177212271i\)
\(L(1)\) \(\approx\) \(0.7624986028 - 1.469218886i\)
\(L(1)\) \(\approx\) \(0.7624986028 - 1.469218886i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (0.286 - 0.957i)T \)
5 \( 1 + (0.998 - 0.0581i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (0.448 - 0.893i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (-0.448 - 0.893i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.993 - 0.116i)T \)
53 \( 1 + (0.0581 + 0.998i)T \)
59 \( 1 + (0.230 - 0.973i)T \)
61 \( 1 + (-0.116 + 0.993i)T \)
67 \( 1 + (0.893 - 0.448i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.286 + 0.957i)T \)
79 \( 1 + (-0.549 + 0.835i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (-0.802 - 0.597i)T \)
97 \( 1 + (-0.549 - 0.835i)T \)
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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

−18.47786800555368665643694828292, −17.82904488181155467851218018103, −17.17172018907916228119400942070, −16.72885043688017509101686946734, −16.11884425738596450288876429503, −15.20100226898451075274014136183, −14.57201075929452734047375426943, −14.29386812925825240863571262416, −13.65403329642460476967328810131, −12.83859115450204477524148693328, −12.04479915754921609409940611132, −10.9953328432713829983040685353, −10.34017508052439795976519276621, −9.67135254941764039617192268877, −8.98366408373515825704660496598, −8.401272175949428118516585801263, −7.51063900475453450561002584088, −6.8924090769193129708221704830, −5.86662755087332062108476605773, −5.26225886670810192185051833500, −4.739626239195349332387190476, −3.950232831258542106243010602863, −3.26717714635456629689072633787, −2.10405143507919288841077067425, −1.31184968835165718712691002115, 0.47514448715743468174968379114, 1.00600856746962140624267595876, 1.84341970750173656977679779339, 2.637011833312725571179098920734, 2.9665309604753406752945802487, 4.15310649807577444885296914676, 5.26675515610319044091091037400, 5.69635105265936698588622819788, 6.21815363951116991339293960360, 7.40719455738786733840906914359, 8.45729232337267417294324298894, 8.60623657530932705317525449285, 9.522242429865238246019828709015, 10.41656774004794406798778490862, 11.096140334466743760262513538100, 11.60669846347432153864948434424, 12.59507250148219667057933110572, 12.95554839994431998860971722955, 13.67325902086569475381720471295, 14.10907981942517002884236188015, 14.82934960921200725792125539721, 15.430443507775418107626527991695, 16.97838601117845962459488455641, 17.27172283407463624045994167826, 18.177951614337678053804124731633

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