Properties

Label 1-4033-4033.1732-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.327 - 0.944i$
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  − 2-s + (0.973 + 0.230i)3-s + 4-s + (0.0581 − 0.998i)5-s + (−0.973 − 0.230i)6-s + (0.893 − 0.448i)7-s − 8-s + (0.893 + 0.448i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.973 + 0.230i)12-s + (0.0581 − 0.998i)13-s + (−0.893 + 0.448i)14-s + (0.286 − 0.957i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.973 + 0.230i)3-s + 4-s + (0.0581 − 0.998i)5-s + (−0.973 − 0.230i)6-s + (0.893 − 0.448i)7-s − 8-s + (0.893 + 0.448i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.973 + 0.230i)12-s + (0.0581 − 0.998i)13-s + (−0.893 + 0.448i)14-s + (0.286 − 0.957i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1732, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.327 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606310248 - 1.142802190i\)
\(L(\frac12)\) \(\approx\) \(1.606310248 - 1.142802190i\)
\(L(1)\) \(\approx\) \(1.118848708 - 0.2885881784i\)
\(L(1)\) \(\approx\) \(1.118848708 - 0.2885881784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.0581 - 0.998i)T \)
13 \( 1 + (0.0581 - 0.998i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.893 + 0.448i)T \)
31 \( 1 + (0.993 + 0.116i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.973 - 0.230i)T \)
53 \( 1 + (-0.686 + 0.727i)T \)
59 \( 1 + (0.286 - 0.957i)T \)
61 \( 1 + (-0.597 - 0.802i)T \)
67 \( 1 + (0.973 - 0.230i)T \)
71 \( 1 + T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (0.286 - 0.957i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (0.835 + 0.549i)T \)
97 \( 1 + (0.993 - 0.116i)T \)
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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

−18.56761784693215373603415318649, −18.209297498599689904268036661378, −17.41734207150424218147921055731, −16.7934753515139444071802705228, −15.56596800700732594249276315811, −15.25812856790783262347507592225, −14.66470577905691094975336357915, −14.11725312894142763358205545968, −13.217606734568085524879406820310, −12.18812481025740181999401745900, −11.638635384094175557787920421428, −10.94736608613898217783467464194, −10.14022182767062698320730004622, −9.46621457253908440561078324156, −8.96120603341361910039343193437, −8.1208763406936666555310015828, −7.52255608374744538001789387312, −6.88841428048878242516328904519, −6.446532616467152573455961585467, −5.134544385733354966843792732427, −4.21662722558064129344901999271, −3.19412912152065551676986539290, −2.25791362685440647299011829815, −2.187593717127567771450481565536, −1.11121040881663983099305735276, 0.71637167759080075815484758849, 1.45889287802264362593380629245, 2.07497215845226258204824935162, 3.28131635861701467635155129796, 3.73854713250344527241001907524, 4.912505425234797656273749675992, 5.56369218555223392936525845915, 6.54516895306687396197636068172, 7.68984738729851731569784212788, 8.06248113577558455337862562181, 8.507846391423318362978877475047, 9.098400139966688443780819463477, 10.004605986427488774056180355777, 10.57487408078839226938876030743, 11.1563288254085621690672589509, 12.19598389290097628147920402729, 12.85994905815728316245029253662, 13.57131295747755034861070361674, 14.37934660836528476280133642442, 15.15727862440588985391667702704, 15.55811663301892145704873342724, 16.50898837095604767451141088996, 17.03738285654279850163824901243, 17.42012524407816549101037239978, 18.65356203125347286197437080413

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