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Properties

Label 1-4033-4033.1910-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.604 + 0.796i$
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 + (0.993 − 0.116i)3-s + (0.766 − 0.642i)4-s + (0.957 − 0.286i)5-s + (−0.893 + 0.448i)6-s + (−0.686 + 0.727i)7-s + (−0.5 + 0.866i)8-s + (0.973 − 0.230i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.686 − 0.727i)12-s + (−0.686 + 0.727i)13-s + (0.396 − 0.918i)14-s + (0.918 − 0.396i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.993 − 0.116i)3-s + (0.766 − 0.642i)4-s + (0.957 − 0.286i)5-s + (−0.893 + 0.448i)6-s + (−0.686 + 0.727i)7-s + (−0.5 + 0.866i)8-s + (0.973 − 0.230i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.686 − 0.727i)12-s + (−0.686 + 0.727i)13-s + (0.396 − 0.918i)14-s + (0.918 − 0.396i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1910, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.869103658 + 0.9284348544i\)
\(L(\frac12)\) \(\approx\) \(1.869103658 + 0.9284348544i\)
\(L(1)\) \(\approx\) \(1.178556460 + 0.2602574461i\)
\(L(1)\) \(\approx\) \(1.178556460 + 0.2602574461i\)

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 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (0.957 - 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.802 + 0.597i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.116 - 0.993i)T \)
31 \( 1 + (0.230 + 0.973i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.998 - 0.0581i)T \)
53 \( 1 + (-0.230 + 0.973i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.957 - 0.286i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (-0.549 + 0.835i)T \)
97 \( 1 + (0.957 - 0.286i)T \)
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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

−18.52225609944876786848269751933, −17.8131128280323191599107531752, −16.89795551888461733217703560607, −16.60292895656566210888105202967, −15.83428438383070103959708043073, −14.80845537296100956204019104343, −14.40360382369737225879880164051, −13.44274499300358736222892941420, −13.06971947565593967860088843619, −12.24615377239651624265074501975, −11.2267644970312242756651776321, −10.47070520642151372592055629787, −9.886990785745545140112658086637, −9.49372467296935082264279442733, −8.83855588853756641011547211618, −8.025966362678727894410931474889, −7.18648824003805564457831796483, −6.789597434478889497444649963505, −5.90907173631777474958139680118, −4.7287361880137705297960491485, −3.582922875724222145790110060161, −3.122314520280650495412102747424, −2.48548012457047564645545559383, −1.52408751759612226165442450504, −0.76280172686590202011972304816, 1.041158953145200546881865866787, 1.9200371610932958381658481287, 2.33483051654251805034034847658, 3.17044351033379842304499087502, 4.37244688224058340089324858942, 5.25943359342231821622584303198, 6.19493542766662314653575950480, 6.82595439714533873989498157204, 7.26044882592785248360872586968, 8.45737537194294131715295258372, 8.98977824716211198433108198162, 9.344120720887431390510300287885, 9.90919510958521143167671875920, 10.61651390601571307876329694579, 11.81784868045137108076445044666, 12.39296997982772695817304719018, 13.202587030488398070871983549006, 13.9240992564649802052041097026, 14.657362818737579353975591882374, 15.22158133158463674973841619640, 15.74796871913878500734506302195, 16.71547232864817278040850280266, 17.299718223285048743466797446930, 17.770057059175078446759013666698, 18.77340040147184615386924572050

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