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Properties

Label 1-4033-4033.343-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.750 - 0.661i$
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  + 2-s + (0.766 + 0.642i)3-s + 4-s + (0.173 − 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.173 + 0.984i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (0.766 + 0.642i)12-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)15-s + 16-s + 17-s + ⋯
L(s)  = 1  + 2-s + (0.766 + 0.642i)3-s + 4-s + (0.173 − 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.173 + 0.984i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (0.766 + 0.642i)12-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)15-s + 16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.750 - 0.661i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.750 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.029120804 - 1.900485863i\)
\(L(\frac12)\) \(\approx\) \(5.029120804 - 1.900485863i\)
\(L(1)\) \(\approx\) \(2.747096275 - 0.3307238067i\)
\(L(1)\) \(\approx\) \(2.747096275 - 0.3307238067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−18.64211339196847262094792424191, −18.263213742436511474336302086665, −17.09843537770070068092997704784, −16.29560015445109592644925668459, −15.54952306151231727524480215118, −14.8701915419356210619157694462, −14.22607583875942628001425677634, −14.06816343549306974005778098741, −13.26314094437452626704880688148, −12.29303008192582784219679275731, −11.96549539850342940569434004586, −11.21854308000672200002528996924, −10.434937643120731806870574775567, −9.50941167609088430071999908882, −8.66889943803062161478881626575, −7.961879693337487447081522809196, −7.15902349404211309982661925275, −6.56321234666622849420177063866, −5.83252346588675386377837243518, −5.380016524859670946920192011854, −3.79799217130414143140424164043, −3.5784568943682993289314687329, −2.74663663658835868453944693953, −1.97221271894135393867912207976, −1.46884694158832981991292888645, 0.87793887347160725873716456640, 1.85023088985397394862663779543, 2.5827402585008903830056556967, 3.596608671798182454104057960923, 4.160424231588032822449322851732, 4.759745234838285315353425566077, 5.30339981922468152995211477545, 6.28126503883540178313257780191, 7.302404525517607015782131987712, 7.885138432718169128671028724398, 8.4723316654868612069656867251, 9.640949103743381049025142092158, 10.08351930484270409364870614175, 10.735624710990066432148418921237, 11.62183333783262258029267320126, 12.55799588923235228004887537572, 13.01602345555772112301355222371, 13.61215461345310182562633306905, 14.28027235018460855343368643090, 14.97730604535936634611752988799, 15.45409312685723188901344169619, 16.27389430118480420928311284040, 16.85322785516012506020557955608, 17.312289967225977969074128942014, 18.449538171087737617334593045283

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