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Properties

Label 1-4033-4033.2198-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.568 + 0.822i$
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  + (−0.5 + 0.866i)2-s + (0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (−0.802 − 0.597i)5-s + (0.0581 + 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.893 − 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (−0.998 − 0.0581i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (−0.802 − 0.597i)5-s + (0.0581 + 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.893 − 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (−0.998 − 0.0581i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.568 + 0.822i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3106200327 + 0.5919774886i\)
\(L(\frac12)\) \(\approx\) \(0.3106200327 + 0.5919774886i\)
\(L(1)\) \(\approx\) \(0.7974649292 + 0.1447081321i\)
\(L(1)\) \(\approx\) \(0.7974649292 + 0.1447081321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.835 - 0.549i)T \)
5 \( 1 + (-0.802 - 0.597i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.918 - 0.396i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.116 + 0.993i)T \)
31 \( 1 + (0.727 + 0.686i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (0.998 + 0.0581i)T \)
53 \( 1 + (-0.998 + 0.0581i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (-0.727 + 0.686i)T \)
67 \( 1 + (-0.448 - 0.893i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (-0.918 + 0.396i)T \)
97 \( 1 + (-0.727 + 0.686i)T \)
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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

−18.66331850611227265201554791161, −17.69836133374789914158600038107, −16.89600568990636255639096740030, −16.16213661883013975918996838354, −15.58888838810393570389521663887, −14.81926972004741595278814863262, −13.85995838928723672239388423393, −13.678892067745396428773873087072, −12.77318601178774097947189601037, −11.75421586995935856444294833166, −11.19737060996810756030416267698, −10.6488496852724813210916044446, −9.94041494995412947480108415945, −9.46714453422332194792574329902, −8.36464251811208681789414868765, −7.91841090687808811190849778561, −7.47365343773453480852920888345, −6.58313705376587930962913760982, −4.98511640166788814526805408225, −4.28454649760739846293055696808, −3.91557638086604094606748831670, −3.0474976518646267021364894214, −2.383165416197828823536262287467, −1.55190132667585824330444034210, −0.2176881051382584763931573379, 1.0189877491583931075051221267, 1.76393018756228138947143565768, 2.79763140936617925039838482296, 3.73139714859552598518588467115, 4.57365532287923435580928171797, 5.5151324834656349100259595374, 6.02682875618771326897452101555, 6.96691765490481592653520751886, 7.94479600292244568648750073450, 8.23874017646551937710501227596, 8.544856678000013173084733224390, 9.30288639171466277967627685720, 10.40771586038585311841365691852, 10.877319581241129255078055215614, 12.234806498467008252827328361544, 12.534346377807059323376654899852, 13.31075090133691659957187218002, 14.09944038455335028334691831998, 14.881353270913833484570293188592, 15.31553828184763632888612824932, 15.83673794024000681214631033750, 16.53413069246839475971832819231, 17.438526396389103642970470935024, 18.22126119739070847564091872942, 18.58174767595862623153179751441

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