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Properties

Label 1-1020-1020.779-r1-0-0
Degree $1$
Conductor $1020$
Sign $0.601 - 0.798i$
Analytic cond. $109.614$
Root an. cond. $109.614$
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  + (0.382 + 0.923i)7-s + (0.923 − 0.382i)11-s − i·13-s + (−0.707 + 0.707i)19-s + (0.923 − 0.382i)23-s + (0.382 − 0.923i)29-s + (−0.923 − 0.382i)31-s + (0.923 + 0.382i)37-s + (−0.382 − 0.923i)41-s + (0.707 + 0.707i)43-s − i·47-s + (−0.707 + 0.707i)49-s + (0.707 − 0.707i)53-s + (−0.707 − 0.707i)59-s + (−0.382 − 0.923i)61-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)7-s + (0.923 − 0.382i)11-s − i·13-s + (−0.707 + 0.707i)19-s + (0.923 − 0.382i)23-s + (0.382 − 0.923i)29-s + (−0.923 − 0.382i)31-s + (0.923 + 0.382i)37-s + (−0.382 − 0.923i)41-s + (0.707 + 0.707i)43-s − i·47-s + (−0.707 + 0.707i)49-s + (0.707 − 0.707i)53-s + (−0.707 − 0.707i)59-s + (−0.382 − 0.923i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.601 - 0.798i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ 0.601 - 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.909316515 - 0.9520462233i\)
\(L(\frac12)\) \(\approx\) \(1.909316515 - 0.9520462233i\)
\(L(1)\) \(\approx\) \(1.175418690 - 0.07436424819i\)
\(L(1)\) \(\approx\) \(1.175418690 - 0.07436424819i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−21.52571280680478044982219972858, −20.761248927936095420694093660868, −19.80485978010905698300708176073, −19.459874440165653898614259350683, −18.33531917740540819662527558511, −17.50456546895446660021714990493, −16.85717976169928736568089267222, −16.25078545506718434436199753975, −15.00143846336014845580888004585, −14.475873385604854361876469427629, −13.66978318625084919239697198379, −12.86058271052365724240113309107, −11.86590428888690567795453188043, −11.11077157442675052027426625609, −10.42767173181749488318607033330, −9.25060613324180310371666724036, −8.813223241256567564343419648950, −7.41322349465902132856065910122, −7.00517397940492212915695784717, −6.031341019619568009219819467113, −4.65773285988256831342238803602, −4.254785372132064921397597067017, −3.11642730485556802289951663054, −1.79392408600031913123500178569, −0.99930724790388141415582221748, 0.50290375022039697902800529823, 1.70596206568161582086173701546, 2.7111673873020275840780560155, 3.70664796693350502445655861752, 4.758449784296464935716308462, 5.74787398197394062478058793655, 6.33679707665102168309294968363, 7.558800497655795897764490250, 8.42793693715681130197237963219, 9.03656071078082224615787818712, 10.03478698679232484255835707343, 10.98291754654241480730612279205, 11.72708375133267615091540041881, 12.52538281238987021026635649882, 13.2583079857402183820409390730, 14.42632410974670954726031437271, 14.9156816741395919253815117708, 15.67487073449829410085851300751, 16.72178013089243324364448849162, 17.32913898708620884758001474222, 18.26708679968426829042262676123, 18.9079523598323754105700187359, 19.68037034422481531897866827963, 20.6026045936235764100694756973, 21.30870584904577800772385630886

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