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Properties

Label 1-4033-4033.31-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.749 + 0.662i$
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  + (−0.642 − 0.766i)2-s + (0.0581 + 0.998i)3-s + (−0.173 + 0.984i)4-s + (−0.802 − 0.597i)5-s + (0.727 − 0.686i)6-s + (0.396 + 0.918i)7-s + (0.866 − 0.5i)8-s + (−0.993 + 0.116i)9-s + (0.0581 + 0.998i)10-s + (−0.0581 + 0.998i)11-s + (−0.993 − 0.116i)12-s + (−0.918 + 0.396i)13-s + (0.448 − 0.893i)14-s + (0.549 − 0.835i)15-s + (−0.939 − 0.342i)16-s + (0.342 − 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.0581 + 0.998i)3-s + (−0.173 + 0.984i)4-s + (−0.802 − 0.597i)5-s + (0.727 − 0.686i)6-s + (0.396 + 0.918i)7-s + (0.866 − 0.5i)8-s + (−0.993 + 0.116i)9-s + (0.0581 + 0.998i)10-s + (−0.0581 + 0.998i)11-s + (−0.993 − 0.116i)12-s + (−0.918 + 0.396i)13-s + (0.448 − 0.893i)14-s + (0.549 − 0.835i)15-s + (−0.939 − 0.342i)16-s + (0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7701027225 + 0.2915071423i\)
\(L(\frac12)\) \(\approx\) \(0.7701027225 + 0.2915071423i\)
\(L(1)\) \(\approx\) \(0.6046166368 + 0.07688053898i\)
\(L(1)\) \(\approx\) \(0.6046166368 + 0.07688053898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (0.0581 + 0.998i)T \)
5 \( 1 + (-0.802 - 0.597i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.0581 + 0.998i)T \)
13 \( 1 + (-0.918 + 0.396i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (-0.549 - 0.835i)T \)
31 \( 1 + (0.918 + 0.396i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.286 + 0.957i)T \)
53 \( 1 + (-0.597 + 0.802i)T \)
59 \( 1 + (0.549 - 0.835i)T \)
61 \( 1 + (-0.957 - 0.286i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (-0.116 - 0.993i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (0.727 - 0.686i)T \)
97 \( 1 + (-0.116 + 0.993i)T \)
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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

−18.21310264372003587518119717294, −17.491534356641306935013394344860, −16.8882502232394993918609527319, −16.54440563136475379712480128357, −15.32322939586892612569766107178, −14.80681468175586205983939227802, −14.36763026721877399883293464665, −13.55283452301379667316743504806, −12.92342173714494483435106402426, −11.974356004042944155974726423400, −11.06832716646716742520920778319, −10.773048982520688335894257806177, −9.994008127703199080708094498397, −8.74613779169916916237739029963, −8.24002974024988588439732062977, −7.78009220759444744041795526124, −6.95748049971944721157965814639, −6.69653846846500487587918802610, −5.73694265029218643187900444340, −4.92015282174121675529591694596, −3.91369780190133209249395878138, −3.013171025569893944158488333162, −2.03130344998348304786896859966, −1.01225378094025623333560801469, −0.39362274487406943087121157028, 0.39237270477281919858070541971, 1.671230668315483705743893525338, 2.48981927008986596528626387877, 3.16453141180633965231535026488, 4.143886226649502101427616400061, 4.84155677919071220802471065852, 5.04112725455924362477804206410, 6.53569430768672416259056602313, 7.61355329498347020771010867279, 8.06600734465960360189931448694, 8.940730655005628444546522886676, 9.356064586054753603964677696660, 9.92246901035751302373017233851, 10.85572373342112138534157295300, 11.49059230652809915508599106083, 12.105056276675654833423277349517, 12.43658840578406814997230498842, 13.45766555169355939717709876653, 14.52422389773545597394550585994, 15.18781243934014448473056069344, 15.6381676527411536594521202885, 16.460894274436901094777737135324, 17.143603851064029034193001815111, 17.46917480857179894077185710355, 18.62276371930259540631517023079

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