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Properties

Label 1-4033-4033.30-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.659 + 0.751i$
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  + (0.984 + 0.173i)2-s + (0.396 + 0.918i)3-s + (0.939 + 0.342i)4-s + (0.686 + 0.727i)5-s + (0.230 + 0.973i)6-s + (0.973 − 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.686 + 0.727i)9-s + (0.549 + 0.835i)10-s + (0.549 − 0.835i)11-s + (0.0581 + 0.998i)12-s + (0.957 + 0.286i)13-s + (0.998 − 0.0581i)14-s + (−0.396 + 0.918i)15-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.396 + 0.918i)3-s + (0.939 + 0.342i)4-s + (0.686 + 0.727i)5-s + (0.230 + 0.973i)6-s + (0.973 − 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.686 + 0.727i)9-s + (0.549 + 0.835i)10-s + (0.549 − 0.835i)11-s + (0.0581 + 0.998i)12-s + (0.957 + 0.286i)13-s + (0.998 − 0.0581i)14-s + (−0.396 + 0.918i)15-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.659 + 0.751i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.970493924 + 8.761834817i\)
\(L(\frac12)\) \(\approx\) \(3.970493924 + 8.761834817i\)
\(L(1)\) \(\approx\) \(2.585507692 + 1.893638815i\)
\(L(1)\) \(\approx\) \(2.585507692 + 1.893638815i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (0.396 + 0.918i)T \)
5 \( 1 + (0.686 + 0.727i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (0.549 - 0.835i)T \)
13 \( 1 + (0.957 + 0.286i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.835 + 0.549i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.957 + 0.286i)T \)
53 \( 1 + (-0.998 + 0.0581i)T \)
59 \( 1 + (-0.918 - 0.396i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (0.448 + 0.893i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (-0.549 + 0.835i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
97 \( 1 + (-0.396 - 0.918i)T \)
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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

−18.15144446122063627469201312360, −17.43648051604984989097425314448, −16.70059946869222944457634114962, −15.908797668339681389357220325770, −15.0070362251588946739163491909, −14.32395615037299542157262486379, −14.07133425996297069158607477550, −13.155497392304297945557736581250, −12.75168132618410597845218860292, −12.01998242452099328848557277869, −11.53564191941990594879049379101, −10.695931593340926646942732193873, −9.59042535167138856701145665983, −9.1080404243035240327566901779, −7.88278053239031292030260682454, −7.74394607649505955402987361163, −6.571916795415365009726410925253, −5.97535450458188873634657797579, −5.3312699324883602256173879916, −4.588062240336305668346328159728, −3.72303304579924325538686293071, −2.78682647601747902672273193619, −1.91994381152030355997525342830, −1.47274068120623484750762426384, −0.7881252131361957507202902904, 1.39107604927907016214499912487, 1.89150714305247959656684593760, 3.08090695219232582581226973793, 3.58065575266466519652748449795, 4.06905421631155897634959721458, 5.280640827092865972173444779801, 5.61115212944513573170997758573, 6.27267388340872806805850561134, 7.44439038477076011472781782143, 7.87974253989550703675954481610, 8.856059008961414509726536053742, 9.60687479960805098263980976743, 10.534910618916590475959532588372, 11.07645263332786067417809415919, 11.41971458094640126273520470983, 12.40574499345765632201907172904, 13.60228601480712985590729718861, 13.9484391655826068669732306817, 14.350442448551921108264841702050, 14.89043524041015937922862991691, 15.72664991816161899193107573603, 16.41001164229161401379439368040, 16.89220916227251020957220009378, 17.73014156305229352409630338089, 18.5662119345523645963387690847

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