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Properties

Label 1-4033-4033.1078-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.795 - 0.606i$
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  − i·2-s + (−0.973 − 0.230i)3-s − 4-s + (−0.998 − 0.0581i)5-s + (−0.230 + 0.973i)6-s + (0.893 − 0.448i)7-s + i·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 + (−0.998 − 0.0581i)13-s + (−0.448 − 0.893i)14-s + (0.957 + 0.286i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (−0.973 − 0.230i)3-s − 4-s + (−0.998 − 0.0581i)5-s + (−0.230 + 0.973i)6-s + (0.893 − 0.448i)7-s + i·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 + (−0.998 − 0.0581i)13-s + (−0.448 − 0.893i)14-s + (0.957 + 0.286i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.795 - 0.606i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1078, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.795 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2905650650 - 0.8608415478i\)
\(L(\frac12)\) \(\approx\) \(0.2905650650 - 0.8608415478i\)
\(L(1)\) \(\approx\) \(0.5518974267 - 0.3529493505i\)
\(L(1)\) \(\approx\) \(0.5518974267 - 0.3529493505i\)

Euler product

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

−18.44167410239381241504635019270, −17.735066753598511887402141959740, −16.994279552321400094176722729722, −16.62917908020474518272571039918, −15.81829572320185824070121733494, −15.309268284359540677397310575481, −14.73805856910615722288019394857, −14.07271213951393855216947858763, −13.031189946788202631247788315269, −12.35695007500696474206196429729, −11.63339213660591969523066147038, −11.18902332273336790320148532255, −10.32254616744115634803591517212, −9.37564772093907738101361998300, −8.77937499035581169192191443448, −7.7585702864383478708515771207, −7.505338034198238268908962822950, −6.66948820912183172754169781814, −5.59281923149104536419327053602, −5.37425177838106721738512261114, −4.55756282164599597996179921323, −3.85585288854631858386613079683, −3.00743815462287356369887622729, −1.35105946450861658135358838158, −0.59121767147638262200064911185, 0.30434760105175895016635936585, 1.06211291716193452954460848023, 1.79044925294523525852434142241, 2.798577339471121110022720296904, 3.83967041809811276969898855554, 4.63417315661600142564287056196, 4.843045431135739576975127307610, 5.73211315945865461806283480717, 7.06517297429489031752380307952, 7.59583300116557676010258322091, 8.058165396913254290184221298501, 9.23563891956230352985159800223, 10.057376420707335344140583096101, 10.53143308046845536869715587789, 11.31599189536144556196784465113, 11.85862992696443999635779560121, 12.36284711717120478221023343116, 12.74516513280966304593914505097, 13.926369450625009296284257527667, 14.528946119653502346435047756688, 15.1711884241198506383671210788, 16.225189545421942938154181288077, 16.89768861281666582947168711228, 17.43915079477637018528276325955, 18.12832792054677019907608044884

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