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Properties

Label 1-4033-4033.2170-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.987 - 0.155i$
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 + (0.993 − 0.116i)3-s + 4-s + (0.727 + 0.686i)5-s + (−0.993 + 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.727 − 0.686i)10-s + (0.727 − 0.686i)11-s + (0.993 − 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (0.802 + 0.597i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.993 − 0.116i)3-s + 4-s + (0.727 + 0.686i)5-s + (−0.993 + 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.727 − 0.686i)10-s + (0.727 − 0.686i)11-s + (0.993 − 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (0.802 + 0.597i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.667109836 - 0.2089706337i\)
\(L(\frac12)\) \(\approx\) \(2.667109836 - 0.2089706337i\)
\(L(1)\) \(\approx\) \(1.427528904 + 0.01703085608i\)
\(L(1)\) \(\approx\) \(1.427528904 + 0.01703085608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (0.727 + 0.686i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (0.727 - 0.686i)T \)
13 \( 1 + (0.686 - 0.727i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.230 - 0.973i)T \)
31 \( 1 + (-0.998 + 0.0581i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.116 - 0.993i)T \)
53 \( 1 + (0.918 + 0.396i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (-0.448 - 0.893i)T \)
67 \( 1 + (0.116 - 0.993i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.973 + 0.230i)T \)
79 \( 1 + (-0.597 + 0.802i)T \)
83 \( 1 + (0.835 + 0.549i)T \)
89 \( 1 + (0.957 - 0.286i)T \)
97 \( 1 + (0.998 + 0.0581i)T \)
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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

−18.31153040591312923111318023719, −17.90808881879075794529450887981, −17.241913020114941513005974589176, −16.4354826773351053730856701455, −16.04261598299216436320762488591, −14.85938229141764790590794465105, −14.66712875940227548260970741054, −13.82239226439216437168759332193, −13.05180071202383852877076176229, −12.29169300883707840109683830747, −11.48009951463469209168165307925, −10.66097929428823266115094168755, −10.04157565662278084828618083616, −9.22548452965899802838841636812, −8.802361268949128550180206664505, −8.37035017790821997587743414559, −7.42247854163408042958323895503, −6.82786336042443119338996682115, −5.99220134710082156520773212853, −4.8397356280189785793297189402, −4.2555286990406845718647715205, −3.23714612545476436575658564773, −2.181262557711771640499225216560, −1.556339275710267389106170691654, −1.23587637751698807714939195206, 0.98916879758863858739074558033, 1.6638466382192646144457605042, 2.34358871825220889603643699003, 3.19855562309625469367728899964, 3.71541153074485991210467731874, 5.23076578858311809372181776407, 5.89739004414942768413774719035, 6.78860020160213412013150536061, 7.46227312636642668395497549452, 8.01374712036887447070153353480, 8.97646930151796551730830464108, 9.12007197422464684959888439411, 10.10448501988606890122337947172, 10.65642874825737131151062206138, 11.524854579609636596859231985656, 11.93076593163089006856333908819, 13.291680826228524480931677085708, 13.78728232659626770860372516442, 14.375292411324840778408487608169, 15.24007783033146981597000675049, 15.50690013258918271929501320265, 16.51358631677457472187708022554, 17.35483057691823263316099655607, 17.94920331434749192749732882046, 18.53382915803659418580222679381

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