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Properties

Label 1-4033-4033.155-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.999 + 0.00620i$
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 + (0.173 − 0.984i)5-s + (0.5 − 0.866i)6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.766 − 0.642i)13-s + (0.5 + 0.866i)14-s + (0.173 + 0.984i)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 + (0.173 − 0.984i)5-s + (0.5 − 0.866i)6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.766 − 0.642i)13-s + (0.5 + 0.866i)14-s + (0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 + 0.00620i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 + 0.00620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001855016988 - 0.5977452648i\)
\(L(\frac12)\) \(\approx\) \(0.001855016988 - 0.5977452648i\)
\(L(1)\) \(\approx\) \(0.5177880219 - 0.1789872175i\)
\(L(1)\) \(\approx\) \(0.5177880219 - 0.1789872175i\)

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 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.954215482308791347611036494123, −18.00556888118852476496497155597, −17.55298563898544284721168634737, −17.22930910034697362524725806985, −16.28632468281389810129775179247, −15.50426617904950051482211577457, −14.8186120806714996685040728500, −14.051175169282213989253530813731, −12.87997779220056706689121898126, −12.433021145475726547413242803084, −11.85408580542793368511201000488, −11.261548704237638986332421210026, −10.56762437485783508098150836798, −9.97508871677656391902600888570, −9.31476714108578282500915194235, −8.418942261708029020460862563910, −7.5686649961667786962420794615, −6.85655163513797252266836195146, −6.43588072646032714519536258244, −5.48425496769737345005945624097, −4.5134204267752017782721456198, −3.72381837611115766683400092465, −2.60803165604428114291953508303, −1.92402708023729735210779867481, −1.4800443603569291143901565511, 0.37462484056449154963900009818, 0.687731614870814659695391524119, 1.62558936066136312617446245990, 2.93745537377978910041568054158, 4.438033172471488567587720705963, 4.61383948962985373592574379516, 5.44358780290771696481259735786, 6.22598117240387430731694451919, 6.83367992619812173676453779675, 7.623696773358057841261772882389, 8.40171497494157628697864204418, 9.178536564973449065281541169643, 9.757319788143202228289324770778, 10.455241563334403418780692519673, 11.127554547714863255972441319648, 11.67071498135164929510886896852, 12.69888704518297948637972844934, 13.33237200233271213772570112376, 14.19132871420739211486876021528, 14.88788860700358936446362998484, 15.7917089899292404589243613105, 16.32909563302191668345309047455, 16.80577722577808211183565937336, 17.352064815933523026385501428777, 17.71314283694256081215482131687

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