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Properties

Label 1-4033-4033.3135-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.999 + 0.00608i$
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 + (0.973 + 0.230i)3-s + (0.173 + 0.984i)4-s + (0.835 + 0.549i)5-s + (0.597 + 0.802i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.0581 + 0.998i)12-s + (−0.0581 + 0.998i)13-s + (−0.686 + 0.727i)14-s + (0.686 + 0.727i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.973 + 0.230i)3-s + (0.173 + 0.984i)4-s + (0.835 + 0.549i)5-s + (0.597 + 0.802i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.0581 + 0.998i)12-s + (−0.0581 + 0.998i)13-s + (−0.686 + 0.727i)14-s + (0.686 + 0.727i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 + 0.00608i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 + 0.00608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01272628300 + 4.180154697i\)
\(L(\frac12)\) \(\approx\) \(0.01272628300 + 4.180154697i\)
\(L(1)\) \(\approx\) \(1.547927708 + 1.751826914i\)
\(L(1)\) \(\approx\) \(1.547927708 + 1.751826914i\)

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

−18.15860196262069002033556337125, −17.56279850701576393942663428824, −16.89186416873189053323207886988, −15.74717163400024898274537549696, −15.196887446256596488787659592243, −14.52692021521274492681501340472, −13.88931657788981045777220001395, −13.23773049697557347680175102523, −12.86128369091654541569909906554, −12.41536305570748929355375309434, −11.11852236415756681262439908764, −10.55827433168837563003793103186, −9.78133910409895398500195368605, −9.31616900140715080889316600616, −8.6054952426029005991440431339, −7.40161474190288161590586851634, −6.94862991691098972881659244468, −6.10829237726961220235663294802, −5.05093276674893648141474664432, −4.528243898407665747174776686187, −3.75903098680701339578053782501, −2.98174912983545462096951901673, −2.072268822590770936853590263909, −1.61296997374222448821376210183, −0.65108458459322769704067130035, 1.93265930486623040838092470765, 2.19980842214208526156525297143, 3.074356492783170718448020239735, 3.78300649751313992826821579242, 4.55881379756331710311423867452, 5.46166928409997300735952423305, 6.24520527622208100616841931481, 6.65349435517536282451006514827, 7.55959482331599845661375279974, 8.508662286726504327464843320926, 9.021955404152504729817495165111, 9.37749574720952284192790744771, 10.74301389901925669374165722797, 11.15352073791771510396809049322, 12.238062693145168817058925427483, 12.9892719838566937365190877708, 13.47007954915511349603354255038, 14.22611731595393121267973542900, 14.63316996970137898572158066748, 15.23122479622367975196751174481, 15.82595781902967860151906788922, 16.78832108301967866414379746894, 17.09673518718426840906110353959, 18.325474922045099111055969148512, 18.75115178996355308975248442471

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