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Properties

Label 1-4033-4033.144-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.169 - 0.985i$
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  + (0.173 − 0.984i)2-s + (−0.286 + 0.957i)3-s + (−0.939 − 0.342i)4-s + (−0.835 + 0.549i)5-s + (0.893 + 0.448i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (0.597 − 0.802i)12-s + (0.893 + 0.448i)13-s + (−0.993 − 0.116i)14-s + (−0.286 − 0.957i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.286 + 0.957i)3-s + (−0.939 − 0.342i)4-s + (−0.835 + 0.549i)5-s + (0.893 + 0.448i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (0.597 − 0.802i)12-s + (0.893 + 0.448i)13-s + (−0.993 − 0.116i)14-s + (−0.286 − 0.957i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7959133193 - 0.9444506520i\)
\(L(\frac12)\) \(\approx\) \(0.7959133193 - 0.9444506520i\)
\(L(1)\) \(\approx\) \(0.8201314182 - 0.3340980610i\)
\(L(1)\) \(\approx\) \(0.8201314182 - 0.3340980610i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.286 + 0.957i)T \)
5 \( 1 + (-0.835 + 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (0.396 - 0.918i)T \)
13 \( 1 + (0.893 + 0.448i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.396 + 0.918i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.835 - 0.549i)T \)
53 \( 1 + (0.597 - 0.802i)T \)
59 \( 1 + (-0.286 - 0.957i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (-0.993 + 0.116i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
97 \( 1 + (-0.686 - 0.727i)T \)
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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

−18.58188595730777757773323046752, −17.77778795523043710334205468073, −17.39826448315615841724819754380, −16.63186267833016205911884830626, −15.79603152290646782990701272568, −15.36014478413966033394205445870, −14.80667945541837731736846879746, −13.74565228436053115183588217907, −13.23124438390598097610150702589, −12.53336869529185764555716664923, −11.93970220931966240496831544959, −11.607757395619105386452552122605, −10.28743643478116063543689129579, −9.28631148282885326170481529470, −8.6351915134394549101088538696, −8.06145922750876596214154806982, −7.522249818837277765624781193417, −6.74409866716084401283102317961, −5.93769858952268237707460731272, −5.476375271463682921343957091123, −4.66233722914235263774074347934, −3.758204707254879192832569781992, −2.956773336225053714578589038885, −1.66714445673757237796079257155, −0.86944962819576659471153133182, 0.521957475880194340603890626618, 1.14778400845373550529849749464, 2.754803787813014575084218485028, 3.42632900617214703061923747583, 3.697970486171130585781579867722, 4.62098859561361095817879520556, 5.10958919160242409425620749610, 6.310555259840617558948553601299, 6.86183791920599569134156297100, 8.190916022555850579164671626081, 8.58426733790689750664821474230, 9.60724758698122754293379232665, 10.10868593491205457794829177698, 10.914924179692240093718916272915, 11.324155665369380011422366871211, 11.69124618080318060283285527012, 12.62700069121904791867129003612, 13.771649181272561668793847161694, 14.06279651713892152596713304341, 14.65040391755282788884307399218, 15.67165981611041642893119876660, 16.29308672747525155726975288139, 16.72595074014819889626670411409, 17.79591808398142929497155523120, 18.38789042440450838585358841435

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