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Properties

Label 1-4033-4033.2711-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.954 - 0.298i$
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 + (0.973 + 0.230i)3-s + (−0.173 − 0.984i)4-s + (−0.835 − 0.549i)5-s + (−0.802 + 0.597i)6-s + (−0.0581 + 0.998i)7-s + (0.866 + 0.5i)8-s + (0.893 + 0.448i)9-s + (0.957 − 0.286i)10-s + (0.957 + 0.286i)11-s + (0.0581 − 0.998i)12-s + (0.998 + 0.0581i)13-s + (−0.727 − 0.686i)14-s + (−0.686 − 0.727i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (0.973 + 0.230i)3-s + (−0.173 − 0.984i)4-s + (−0.835 − 0.549i)5-s + (−0.802 + 0.597i)6-s + (−0.0581 + 0.998i)7-s + (0.866 + 0.5i)8-s + (0.893 + 0.448i)9-s + (0.957 − 0.286i)10-s + (0.957 + 0.286i)11-s + (0.0581 − 0.998i)12-s + (0.998 + 0.0581i)13-s + (−0.727 − 0.686i)14-s + (−0.686 − 0.727i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.954 - 0.298i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.954 - 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2093472660 + 1.369281904i\)
\(L(\frac12)\) \(\approx\) \(-0.2093472660 + 1.369281904i\)
\(L(1)\) \(\approx\) \(0.8403983319 + 0.5100718671i\)
\(L(1)\) \(\approx\) \(0.8403983319 + 0.5100718671i\)

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 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (0.957 + 0.286i)T \)
13 \( 1 + (0.998 + 0.0581i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.973 - 0.230i)T \)
31 \( 1 + (-0.893 - 0.448i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.116 - 0.993i)T \)
53 \( 1 + (0.448 + 0.893i)T \)
59 \( 1 + (-0.727 + 0.686i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (0.549 + 0.835i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.998 + 0.0581i)T \)
83 \( 1 + (-0.973 - 0.230i)T \)
89 \( 1 + (0.396 - 0.918i)T \)
97 \( 1 + (-0.835 - 0.549i)T \)
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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

−18.08934428470812433038029475177, −17.46837831121401086150966951647, −16.59941132099930868186449488979, −15.97292498816775478474353397937, −15.18930158751868240127105289331, −14.35426303581187076760149655856, −13.834277962936978104578524129464, −12.99345729276050829539885586780, −12.56411946428790965404208396312, −11.526622887507967981961896291934, −10.787180609540170201062856967036, −10.66300131931385026512023528537, −9.38740310265752990662675878400, −8.96779445036844839135424513468, −8.24076640326246850067263064195, −7.5858816677685628958007233380, −6.94442102682699726689452955986, −6.415854073500048075835486919800, −4.59456805940528868132642796415, −3.8513314774411485036758662037, −3.58194660010156090062688053573, −2.838654875959888823734443117467, −1.760247513377789792393433839269, −1.06005446247662373651570059941, −0.240396756119160475556475218221, 1.15358239299608923456758119366, 1.666558215667702438622064499004, 2.81010242171725265632651293820, 3.82240507787336193425317059672, 4.35311600404774478673205032501, 5.37110128551832768208366690183, 5.9628620240526470584537638507, 7.20711819617552021882086715474, 7.4396358000375173809593633805, 8.47433089272106558693538262683, 8.84810150463295598116415461094, 9.31858358959941258321583730669, 9.97133741055196460726648126011, 11.205400113408826258277527640789, 11.60135577738888413258075152037, 12.644013781345333862411406479962, 13.37995660836848703436817118948, 14.122184677060403430869127373225, 14.96071633676470765061343573807, 15.21253814464995640484089566932, 16.0082238037332297882413643231, 16.35378023929694761175268449529, 17.15566398370446938254428735397, 18.28254509911583045231930369611, 18.63094995289960438791094142002

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