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Properties

Label 1-4033-4033.346-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.982 - 0.184i$
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  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (0.5 − 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)14-s + (0.642 − 0.766i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (0.5 − 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)14-s + (0.642 − 0.766i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.982 - 0.184i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.982 - 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3433504941 - 3.683442063i\)
\(L(\frac12)\) \(\approx\) \(0.3433504941 - 3.683442063i\)
\(L(1)\) \(\approx\) \(1.539032949 - 1.446715984i\)
\(L(1)\) \(\approx\) \(1.539032949 - 1.446715984i\)

Euler product

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

−18.747306279835042588222549009267, −18.08376240485787404462113133669, −17.55273113408105860427577701783, −16.46915356994346788846966483052, −15.89952509817431328277549773829, −15.157027426948771789109393441217, −14.68689726975731385728093387652, −14.07371293908916141343357472872, −13.533184117937462234674715314626, −12.75359558369176349416270653050, −12.29790577049459482270796174028, −11.48553443904257635246199346320, −10.32253674965883803089723867715, −9.7392275829727089814839209924, −8.926356433674628126507576799194, −8.231627185483805009419199724132, −7.384735625405758813045285483, −6.8101821128398138011939055152, −6.2177489128713505882395637347, −5.73020069452700411404510935260, −4.17122870820505094009357789502, −3.85506932064431732054454034412, −3.118183703352200183312321998532, −2.231255497215124345103894763644, −1.74760935582572689997919098343, 0.65069662154835534298479620668, 1.29733922994102439785171603234, 2.54284428406281359144595781768, 2.95564583844518613343430157737, 3.88695892404196291629289131749, 4.29398403573935129509976809836, 5.311344151784753680616334368190, 5.906088014856001362300515522080, 6.772189315645902308913769501447, 7.71593140833943661235858731365, 8.841507600828739473673908203314, 9.10248805796226063010447000012, 9.831300622519501903148887318007, 10.54777618226822431772070735704, 11.21448270844154674685698617585, 12.24151820833482550597647865216, 12.84374603983921621416916350766, 13.507680717021440742718845812294, 13.80503244826180584039180408419, 14.40475607284647004021811412715, 15.61195393776205175482335653939, 16.002182701797523780730765728635, 16.27760287403193785739156051407, 17.5538970730731959726328947990, 18.37788996213125094337615648051

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