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Properties

Label 1-4033-4033.3822-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.229 - 0.973i$
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 + (0.893 − 0.448i)3-s + (0.766 − 0.642i)4-s + (−0.396 + 0.918i)5-s + (0.686 − 0.727i)6-s + (−0.993 + 0.116i)7-s + (0.5 − 0.866i)8-s + (0.597 − 0.802i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.396 − 0.918i)12-s + (0.993 − 0.116i)13-s + (−0.893 + 0.448i)14-s + (0.0581 + 0.998i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.893 − 0.448i)3-s + (0.766 − 0.642i)4-s + (−0.396 + 0.918i)5-s + (0.686 − 0.727i)6-s + (−0.993 + 0.116i)7-s + (0.5 − 0.866i)8-s + (0.597 − 0.802i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.396 − 0.918i)12-s + (0.993 − 0.116i)13-s + (−0.893 + 0.448i)14-s + (0.0581 + 0.998i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3822, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.426421231 - 3.065996881i\)
\(L(\frac12)\) \(\approx\) \(2.426421231 - 3.065996881i\)
\(L(1)\) \(\approx\) \(2.007078626 - 0.9375064134i\)
\(L(1)\) \(\approx\) \(2.007078626 - 0.9375064134i\)

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 \( 1 + (0.893 - 0.448i)T \)
5 \( 1 + (-0.396 + 0.918i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (-0.0581 - 0.998i)T \)
13 \( 1 + (0.993 - 0.116i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (-0.396 - 0.918i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (-0.993 + 0.116i)T \)
59 \( 1 + (0.0581 + 0.998i)T \)
61 \( 1 + (0.686 - 0.727i)T \)
67 \( 1 + (0.597 - 0.802i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (-0.0581 + 0.998i)T \)
89 \( 1 + (0.286 + 0.957i)T \)
97 \( 1 + (0.993 - 0.116i)T \)
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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

−18.87854002560135058111692192699, −17.87308075956702374105236381349, −16.98773290852402407394544746533, −16.104548076148983362537874978529, −15.84405723174568313453417467653, −15.56367370354394519075301538347, −14.49946619443571456022304571762, −13.8256321684400768356420727242, −13.26803107230998472490318772613, −12.80421684826340617908121418776, −11.95747629473648366287198744407, −11.395833967919021651649463055787, −10.17506862250456588481039737987, −9.66307936562585448531803304054, −8.83005223819389099802194536320, −8.2080263562871979105167854358, −7.32418343669003584007996701494, −6.92864813049301533647954774210, −5.759430890270758231438160645735, −5.0259286748895894736534863832, −4.417790864396707526229951627632, −3.56073275221096507891306857201, −3.27296697084467799392842857687, −2.170727099362742287210301953577, −1.291028637279293156408629126806, 0.663934330839809815719623493990, 1.72072109248829053477002543562, 2.70405462167011324261036700584, 3.25066658809970667870625257792, 3.57342628752363834301920698607, 4.43649541641634715232410054550, 5.803892249183500239109396803796, 6.38592656829316786462898877474, 6.73714050981363191241813574377, 7.731666469622150180733634186275, 8.41684898978958412288068109853, 9.31472861083845210760207278447, 10.18278956045703403043724778836, 10.75460019814779760547474774355, 11.50268180900962592098660389929, 12.2626655245493357849595297516, 12.99046128214442250155968483612, 13.52383197249772370813957832953, 14.04747830364945381389504831088, 14.75426707773368969656287680987, 15.40060496843124070197185366294, 15.93916991333602074835805925361, 16.53599970400964410019046406691, 17.99561070796591222326635030222, 18.63516395418903920517905327950

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