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Properties

Label 1-1012-1012.35-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.487 + 0.873i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.564 − 0.825i)3-s + (−0.362 + 0.931i)5-s + (−0.985 + 0.170i)7-s + (−0.362 − 0.931i)9-s + (0.466 − 0.884i)13-s + (0.564 + 0.825i)15-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (−0.415 + 0.909i)21-s + (−0.736 − 0.676i)25-s + (−0.974 − 0.226i)27-s + (0.921 + 0.389i)29-s + (−0.941 − 0.336i)31-s + (0.198 − 0.980i)35-s + (−0.254 + 0.967i)37-s + ⋯
L(s)  = 1  + (0.564 − 0.825i)3-s + (−0.362 + 0.931i)5-s + (−0.985 + 0.170i)7-s + (−0.362 − 0.931i)9-s + (0.466 − 0.884i)13-s + (0.564 + 0.825i)15-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (−0.415 + 0.909i)21-s + (−0.736 − 0.676i)25-s + (−0.974 − 0.226i)27-s + (0.921 + 0.389i)29-s + (−0.941 − 0.336i)31-s + (0.198 − 0.980i)35-s + (−0.254 + 0.967i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.487 + 0.873i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.487 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2530757321 + 0.4309090952i\)
\(L(\frac12)\) \(\approx\) \(0.2530757321 + 0.4309090952i\)
\(L(1)\) \(\approx\) \(0.8485359068 + 0.01403418356i\)
\(L(1)\) \(\approx\) \(0.8485359068 + 0.01403418356i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.564 - 0.825i)T \)
5 \( 1 + (-0.362 + 0.931i)T \)
7 \( 1 + (-0.985 + 0.170i)T \)
13 \( 1 + (0.466 - 0.884i)T \)
17 \( 1 + (-0.0855 + 0.996i)T \)
19 \( 1 + (-0.921 + 0.389i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (-0.941 - 0.336i)T \)
37 \( 1 + (-0.254 + 0.967i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.696 - 0.717i)T \)
59 \( 1 + (-0.897 + 0.441i)T \)
61 \( 1 + (-0.610 + 0.791i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (-0.198 - 0.980i)T \)
73 \( 1 + (-0.516 + 0.856i)T \)
79 \( 1 + (-0.466 + 0.884i)T \)
83 \( 1 + (-0.998 + 0.0570i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.998 - 0.0570i)T \)
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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

−21.315032009408649659421193594203, −20.57477991818270537978088001070, −19.787269068615417771147732603429, −19.39609799522260515118936932491, −18.413305682173097481844378688, −17.06527460455737036749735990694, −16.500242546882756280325481271778, −15.89208058759743809604858542152, −15.342601970880459894853254595925, −14.14642601832743285214363046531, −13.52795223589615076772010492285, −12.71823270663783032109162459780, −11.78263454403348585498822766403, −10.85479205013686431434333844675, −9.94410899913166937467084286189, −9.00818076965667039139733662360, −8.84392980200391963065667026921, −7.62294268475210702545973214520, −6.659464999173245584188732908114, −5.50152761424362623138663308467, −4.54284757796515805227553418422, −3.929495144972907297016047435707, −3.00437576309939101689441797334, −1.84962406825292061010457025219, −0.188499966492187158501626113850, 1.44502357752827342070362061302, 2.659752605063707441487420856890, 3.24762015983851260070021151059, 4.06993734808373318467646716353, 5.866505097564758805588778361709, 6.38664123733507391012696063031, 7.14654888648608584872087426757, 8.114816403514617147052516813287, 8.69096701481287826586758256786, 9.9265844615079339052320951728, 10.57631155893717004160021354970, 11.61030532133922825660151683987, 12.58879526784504286949588694177, 13.04326903832777893228041204487, 13.936362606859627573244133071749, 14.950838917328924308363816811924, 15.23442379414089766387525877958, 16.337695498320791253201397998227, 17.38327384822949537171830855282, 18.270138668998042722304477660809, 18.74473200793382425680795534920, 19.61932474031421651481850109265, 19.88723772030782507145912352381, 21.08570447852031404317464267253, 21.98876318259781670847710905165

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