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Properties

Label 1-1021-1021.652-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.826 + 0.562i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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 + 3-s + 4-s + (−0.5 + 0.866i)5-s + 6-s + 7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + (−0.5 + 0.866i)5-s + 6-s + 7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.826 + 0.562i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (652, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.826 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.184686056 + 1.288103532i\)
\(L(\frac12)\) \(\approx\) \(4.184686056 + 1.288103532i\)
\(L(1)\) \(\approx\) \(2.695609455 + 0.4630356342i\)
\(L(1)\) \(\approx\) \(2.695609455 + 0.4630356342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−21.38809984033533964172440557051, −20.8661884805677068376320446828, −20.11804741309705184505434197163, −19.57419374836963906492553715616, −18.66961185544786359880758325639, −17.53847609396121218820925340775, −16.4953106750693678359105866177, −15.55021659792530484390282479039, −15.4232002550051166960598898082, −14.22205112368709620891916073791, −13.68109560743551918796657611946, −12.95366525962821322293858671555, −12.273783043294234486997877697752, −11.02757675677000957852851592250, −10.79784042242639694560207605224, −9.1017037694226813723145816062, −8.40827088133565990774639336195, −7.86365714789941406498656721274, −6.863571446730654331513264539710, −5.604714962367107833286271595584, −4.86283186988375182118647112698, −3.909338778412645662830053890278, −3.402654660951220935387765562663, −2.02932508906203277654331725201, −1.35301936701532352964784057353, 1.729923716857436934096100472193, 2.34191816540868094371773516885, 3.32887192709234945012331430515, 4.18077156851842923571217471702, 4.78562299774969997066493139360, 6.1728238343563064554029232926, 7.01089442140961012230202689111, 7.86821273875400906892282972699, 8.28775375477437684754472483909, 9.830330531430959621641473579623, 10.61179323647943747543868457979, 11.39749785438855450972543852122, 12.1788433922080890014285892947, 13.24768096999120678017412360818, 13.82018895730960836221488350485, 14.72085922268700401382293082637, 15.04691164021882647443193664840, 15.72092547309025868755199476499, 16.684480638902290656762116107089, 18.291274835845151141449483368172, 18.394460105703857593591152226295, 19.61008352003229970062414369216, 20.440756003344123353493563156317, 20.782389412086253732348877892494, 21.54221731313615741685713877135

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