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Properties

Label 1-4033-4033.638-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.302 + 0.953i$
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  + (−0.173 + 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (0.766 − 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 0.342i)14-s + 15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (0.766 − 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 0.342i)14-s + 15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.302 + 0.953i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.302 + 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7492554374 + 0.5482085581i\)
\(L(\frac12)\) \(\approx\) \(0.7492554374 + 0.5482085581i\)
\(L(1)\) \(\approx\) \(0.6452549273 + 0.2946967242i\)
\(L(1)\) \(\approx\) \(0.6452549273 + 0.2946967242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−18.601519970607906904071624848477, −17.93775605628246977384062408079, −17.09267350261482820539434184965, −16.63538620770339889146194772792, −15.58304327515363569987156315900, −14.777441697279033312611850544783, −13.938768619787203869588222768018, −13.53860779563729019645803894253, −12.53771601754167981519839461299, −11.98992503529715097697114109023, −11.644787110672862110251066888880, −10.90864570406402896953006839817, −10.33708673128051456890011248504, −9.20360190250543225064485416718, −8.78289724975111053277226586440, −7.91685422526762075737675056434, −7.08002640859293676921269040703, −6.37967340636165045586073439373, −5.74759134896286328138211657338, −4.742727859452193738143134970450, −3.666624322555253936922809566, −3.21776813254726716575186323043, −2.215132997952266917735987516177, −1.65418400875120580134993147626, −0.52372307254064493178028424220, 0.69047049800057512572020708688, 1.22714101790566189476033204346, 3.32620168636248236261948205058, 3.81941167776617217141035235458, 4.506116913229702432148270556465, 5.110612450250424770767151203839, 5.936074812824385571170971951243, 6.58495262027263562512654414940, 7.369601773662449597098158189747, 8.27063436907855973069921172972, 8.782999525002909265844701568491, 9.61822469401715298005673073966, 10.11727135184220006344318929451, 10.81179937463379047701673988202, 11.764673285804183252742467164360, 12.70201972681122053985910460948, 12.9618623341350343467880428470, 14.20796101842975270070309573922, 14.63274523930572214347549489058, 15.37748937031792373971503635686, 16.157484185889385113966287061693, 16.48842198346886526329105251998, 17.015861645619830527659297514673, 17.496270296120542656185675987105, 18.445176530559753083094673931047

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