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Properties

Label 1-4033-4033.2460-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.704 + 0.709i$
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  + (0.939 − 0.342i)2-s + (−0.973 + 0.230i)3-s + (0.766 − 0.642i)4-s + (0.448 − 0.893i)5-s + (−0.835 + 0.549i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.116 − 0.993i)10-s + (−0.116 − 0.993i)11-s + (−0.597 + 0.802i)12-s + (0.0581 + 0.998i)13-s + (−0.597 + 0.802i)14-s + (−0.230 + 0.973i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.973 + 0.230i)3-s + (0.766 − 0.642i)4-s + (0.448 − 0.893i)5-s + (−0.835 + 0.549i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.116 − 0.993i)10-s + (−0.116 − 0.993i)11-s + (−0.597 + 0.802i)12-s + (0.0581 + 0.998i)13-s + (−0.597 + 0.802i)14-s + (−0.230 + 0.973i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.704 + 0.709i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1633381438 - 0.3921583533i\)
\(L(\frac12)\) \(\approx\) \(-0.1633381438 - 0.3921583533i\)
\(L(1)\) \(\approx\) \(1.019442792 - 0.4371362549i\)
\(L(1)\) \(\approx\) \(1.019442792 - 0.4371362549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.116 - 0.993i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.116 + 0.993i)T \)
31 \( 1 + (-0.957 + 0.286i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.998 - 0.0581i)T \)
53 \( 1 + (0.802 + 0.597i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (-0.116 + 0.993i)T \)
67 \( 1 + (0.918 + 0.396i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (-0.957 - 0.286i)T \)
97 \( 1 + (-0.230 - 0.973i)T \)
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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

−18.675162719454009809980367844315, −18.013815072897983248946214878873, −17.44840068459644589673970091610, −16.86349918695654562670043022418, −16.12551220503914056897319746587, −15.27970118159992606185550534595, −15.1157684567647051014547408356, −13.861577360884747549127370975180, −13.41445255388263834762102627080, −12.87713835101192136451175860389, −12.17785215154828918609993640679, −11.380297015589082509897474428015, −10.810470206619587483721976917390, −10.03791948409679229154558009485, −9.61128787113939656896254296451, −7.91806050619328463852285349567, −7.413540036155900029184545417099, −6.74501133601927908775688753806, −6.30768698526157513274324254058, −5.49596381387059913799222329402, −4.953040749483463255804764680042, −3.86850068276390491097029515853, −3.339132067130776367494231821747, −2.30494807606881569831249813567, −1.57233796562168544974786547374, 0.09039475720733584520628615918, 1.19372786618056155396879849462, 2.01233905418542615342328792925, 3.00975019150752011437944470304, 3.879190364409062589997494488290, 4.647608898380183908268259052601, 5.272303602521641974112467190715, 5.82580616282145627259041688624, 6.63846801060253282739662622989, 6.90396130675946273322994803754, 8.58673379459978689304348415715, 9.140029554539091797487213271584, 9.864174297614688412365333881, 10.67671576403487986956424725776, 11.32670002101279622696978088547, 11.9158107541436439850928384486, 12.63780462187735902754859771753, 13.12366812739929364299795329017, 13.65335104131444019507368125052, 14.58979546366148801575707969390, 15.5515607093946816165903051988, 15.99741859390697390248378963347, 16.59194688959140484712495106618, 16.97068767253209490210634776966, 18.29556798380132126229968563317

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