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Properties

Label 1-4033-4033.103-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.334 - 0.942i$
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 + (−0.597 − 0.802i)3-s + (0.173 + 0.984i)4-s + (0.727 + 0.686i)5-s + (−0.0581 + 0.998i)6-s + (0.973 + 0.230i)7-s + (0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (0.686 − 0.727i)12-s + (−0.973 − 0.230i)13-s + (−0.597 − 0.802i)14-s + (0.116 − 0.993i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.597 − 0.802i)3-s + (0.173 + 0.984i)4-s + (0.727 + 0.686i)5-s + (−0.0581 + 0.998i)6-s + (0.973 + 0.230i)7-s + (0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (0.686 − 0.727i)12-s + (−0.973 − 0.230i)13-s + (−0.597 − 0.802i)14-s + (0.116 − 0.993i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.334 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7143129837 - 0.5045833117i\)
\(L(\frac12)\) \(\approx\) \(0.7143129837 - 0.5045833117i\)
\(L(1)\) \(\approx\) \(0.6268243929 - 0.2521059492i\)
\(L(1)\) \(\approx\) \(0.6268243929 - 0.2521059492i\)

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 \( 1 + (-0.597 - 0.802i)T \)
5 \( 1 + (0.727 + 0.686i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (0.116 - 0.993i)T \)
13 \( 1 + (-0.973 - 0.230i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (-0.727 + 0.686i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (-0.230 + 0.973i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (0.998 + 0.0581i)T \)
67 \( 1 + (-0.957 - 0.286i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.993 + 0.116i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (0.549 - 0.835i)T \)
97 \( 1 + (0.230 - 0.973i)T \)
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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

−17.99518737245780719032607860749, −17.78390643540853280173109504080, −17.30746445973549694221520999427, −16.63923685097464313783993406839, −16.15126824841365992925959911035, −15.2641496203592014492281905686, −14.59790199786047443766544915403, −14.31511117461870818583771504317, −13.16438006475648416280297801688, −12.28071712516978442980488472433, −11.56048673453610041987242075492, −10.78089982414241815406732831437, −10.20805646528688813695863091761, −9.38778806845842977352659599908, −9.22300332706680299347533777507, −8.1960673189918396030455827467, −7.45306586372031962688482218854, −6.65553190215254488537767017065, −5.81227878280259575048752189612, −5.2331612722142787166285585876, −4.56352228044183728169532259304, −4.0982932295462461942102903618, −2.08997597591923607662089599244, −1.93767424857464771916551090509, −0.62510027141769901798537617050, 0.52378318423143674888245109939, 1.63337038295397243895745776149, 2.24437821217729888149174134935, 2.67278911340101640221132317386, 3.94260641232052728893848318554, 4.90862339941208171617191207213, 5.80694469543355289682501783156, 6.47329797582430440558648664621, 7.23966850054989051072330690201, 7.87001584367424129893628727768, 8.620264662677922621028581464049, 9.27613113863849228857594512669, 10.35543580867062514187433891783, 10.833278897223070935689313116470, 11.31725347554721177246451784427, 11.93658635189520599879287313159, 12.84195981433726151878428849352, 13.30171583905358591344982460423, 14.15564855989674065205931948065, 14.77439371196978084930170668500, 15.821567391754596909349532016777, 16.76733721909249892574270006523, 17.23728724417213151532699680344, 17.86818446184819439740376639608, 18.13620747027488975874184666675

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