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Properties

Label 1-1020-1020.539-r1-0-0
Degree $1$
Conductor $1020$
Sign $-0.806 - 0.591i$
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  + (0.923 − 0.382i)7-s + (−0.382 − 0.923i)11-s − i·13-s + (0.707 − 0.707i)19-s + (−0.382 − 0.923i)23-s + (0.923 + 0.382i)29-s + (0.382 − 0.923i)31-s + (−0.382 + 0.923i)37-s + (−0.923 + 0.382i)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 + (−0.923 + 0.382i)61-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)7-s + (−0.382 − 0.923i)11-s − i·13-s + (0.707 − 0.707i)19-s + (−0.382 − 0.923i)23-s + (0.923 + 0.382i)29-s + (0.382 − 0.923i)31-s + (−0.382 + 0.923i)37-s + (−0.923 + 0.382i)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 + (−0.923 + 0.382i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.806 - 0.591i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ -0.806 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5049149511 - 1.541168964i\)
\(L(\frac12)\) \(\approx\) \(0.5049149511 - 1.541168964i\)
\(L(1)\) \(\approx\) \(1.046804118 - 0.3370492601i\)
\(L(1)\) \(\approx\) \(1.046804118 - 0.3370492601i\)

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 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
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 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.923 + 0.382i)T \)
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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

−21.50506556013666760881681703249, −21.072078423115689044864218722318, −20.24881106395746077614441807617, −19.3719016037825595202760296562, −18.50581838998079930030241156080, −17.799910503479900386286610998444, −17.239414096874654515810743510758, −16.04661309152033265254329127947, −15.546775579893703771944148575425, −14.42522888036039199659647713848, −14.09453051422163966003257895181, −12.945469386114417180538664124342, −11.95348422784128971418202925717, −11.60021467405890381604595669888, −10.43822073764186890795758787991, −9.68529034164189098840884621446, −8.76653529656244849980322681238, −7.88404025334314869618758930507, −7.16871070230149816761676776839, −6.09623386992822216230714218488, −5.07133243284592217539164883834, −4.46937588082458776086175341277, −3.288111274632793611109322665694, −2.04304260206993049428455336719, −1.422674282600032737325427453229, 0.34063117043088055466434961684, 1.23108969337378045005181731021, 2.56678357679647780824923482702, 3.40590617545074757089247733528, 4.623647018923832042464256804631, 5.289778770434026517956002783213, 6.27784235894145605848333637498, 7.3425301238889902407985256239, 8.19211174839332539637848014524, 8.69129554350082560444157367698, 10.071137085418623880614776842084, 10.625888832326645382343738269135, 11.485866974522561357726633470300, 12.22533645029792189214110804011, 13.47916153215723606828517278620, 13.74837262078371741201301387385, 14.87956078559232122737885278662, 15.49426441817777081463115499221, 16.4744716957129217799033931484, 17.18842836780556022743991057543, 18.087011385082646696707527269184, 18.538562505485250033108651627007, 19.704971656079788269382670297142, 20.35845227524255005509502071312, 21.01121602489731062838211470031

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