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Properties

Label 1-4033-4033.1175-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.891 - 0.452i$
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 + (0.893 − 0.448i)3-s + (−0.173 + 0.984i)4-s + (−0.597 − 0.802i)5-s + (0.918 + 0.396i)6-s + (0.396 − 0.918i)7-s + (−0.866 + 0.5i)8-s + (0.597 − 0.802i)9-s + (0.230 − 0.973i)10-s + (0.230 + 0.973i)11-s + (0.286 + 0.957i)12-s + (0.116 + 0.993i)13-s + (0.957 − 0.286i)14-s + (−0.893 − 0.448i)15-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.893 − 0.448i)3-s + (−0.173 + 0.984i)4-s + (−0.597 − 0.802i)5-s + (0.918 + 0.396i)6-s + (0.396 − 0.918i)7-s + (−0.866 + 0.5i)8-s + (0.597 − 0.802i)9-s + (0.230 − 0.973i)10-s + (0.230 + 0.973i)11-s + (0.286 + 0.957i)12-s + (0.116 + 0.993i)13-s + (0.957 − 0.286i)14-s + (−0.893 − 0.448i)15-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.891 - 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08454136620 + 0.3536194183i\)
\(L(\frac12)\) \(\approx\) \(-0.08454136620 + 0.3536194183i\)
\(L(1)\) \(\approx\) \(1.515088737 + 0.3843905613i\)
\(L(1)\) \(\approx\) \(1.515088737 + 0.3843905613i\)

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 \( 1 + (0.893 - 0.448i)T \)
5 \( 1 + (-0.597 - 0.802i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.230 + 0.973i)T \)
13 \( 1 + (0.116 + 0.993i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.973 + 0.230i)T \)
31 \( 1 + (-0.835 + 0.549i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.116 + 0.993i)T \)
53 \( 1 + (-0.957 + 0.286i)T \)
59 \( 1 + (0.448 - 0.893i)T \)
61 \( 1 + (0.973 + 0.230i)T \)
67 \( 1 + (0.727 - 0.686i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (-0.230 - 0.973i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (0.835 + 0.549i)T \)
97 \( 1 + (-0.893 + 0.448i)T \)
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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

−18.37265560496821696170207030286, −17.4048621951205638858786809135, −16.03991424322236404101609345018, −15.641536640080494632670093001375, −14.96585448880927675383372197673, −14.594932679426151296447290959884, −13.85455822575559104211857796822, −13.236927940237494711719118562239, −12.425079355004603549472917338576, −11.505909312505918359660217582335, −11.22507885141578042332447882621, −10.34376683386094854016291378008, −9.812419648491019068716048245759, −8.78457646827404813824895144804, −8.426539541968711814797172085503, −7.47570293159188089086817060577, −6.57390464013080410432405064438, −5.53271760931096045514102748020, −5.112065668864251776221577673546, −3.98042214084230090331365176482, −3.46966892092779095456385293426, −2.77620271623341708519928127641, −2.33773588079524963965710174384, −1.17141039278270541858650072439, −0.03368055569346053907516613370, 1.31229341623458636035573833606, 1.87440604689875957347095771129, 3.16241304879945919029143475633, 3.892474597262980286905136114500, 4.45290003956264988694560790816, 4.89058844095926228944875067768, 6.33108102751948637214654840279, 6.7898393694360023474887231170, 7.56342413803070526890563193701, 8.11785923172198381273726814140, 8.635318538666583974454085840168, 9.38705959452489774677333426469, 10.31297913135425083345092487607, 11.430497830268111096074020013997, 12.23834584470610768580715418468, 12.68881077880496873143248670478, 13.20393680233052779003932978070, 14.12092780073013688907550386688, 14.63259884235619120108435812964, 14.9571315119137358381905770450, 16.021848046941476234818981105403, 16.52720518893097603398414288044, 17.187688232015158765623949023863, 17.79718286491907028974373139015, 18.75675833262519056433196416677

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