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Properties

Label 1-2240-2240.1693-r0-0-0
Degree $1$
Conductor $2240$
Sign $-0.256 - 0.966i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
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)3-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.923 + 0.382i)13-s − 17-s + (0.382 − 0.923i)19-s + (0.707 − 0.707i)23-s + (−0.923 + 0.382i)27-s + (0.923 + 0.382i)29-s + 31-s − i·33-s + (0.923 − 0.382i)37-s + (0.707 − 0.707i)39-s + (0.707 + 0.707i)41-s + (0.382 + 0.923i)43-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.923 + 0.382i)13-s − 17-s + (0.382 − 0.923i)19-s + (0.707 − 0.707i)23-s + (−0.923 + 0.382i)27-s + (0.923 + 0.382i)29-s + 31-s − i·33-s + (0.923 − 0.382i)37-s + (0.707 − 0.707i)39-s + (0.707 + 0.707i)41-s + (0.382 + 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ -0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9844417264 - 1.279304271i\)
\(L(\frac12)\) \(\approx\) \(0.9844417264 - 1.279304271i\)
\(L(1)\) \(\approx\) \(1.059137509 - 0.4454453027i\)
\(L(1)\) \(\approx\) \(1.059137509 - 0.4454453027i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 - T \)
19 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.382 - 0.923i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 + (0.382 - 0.923i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 - iT \)
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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

−20.06190740212539540845108713259, −19.2529880409545784917808878246, −18.52253651400575807530217583128, −17.71014933254234154717622874272, −16.95448146259390376798321595227, −16.02580064343451815562046833577, −15.65992941828393845409744687084, −15.08725895554602121292717746041, −13.9929586466524028231912159748, −13.59009620733131994492996052636, −12.80011132806801766165693397276, −11.66180252510758862705761959609, −10.96917677520546808943544952875, −10.39138232659611960871391719200, −9.66587409828683562915276557604, −8.754291957953769294775917357194, −8.23863783413253377141167744278, −7.4668040645676772841184353433, −6.20094798109583856764742043356, −5.57667538492084972388290969629, −4.70474166715195974060676779166, −3.93193445367551949490538937488, −3.066185623292558572867978798682, −2.43229020107451394975435525448, −1.07160794199600488252716923845, 0.56704833028349998923007453619, 1.58566815641046669059892798370, 2.57515322185811368456576106990, 3.07551158197921088584584100529, 4.379843052671305583849957611400, 5.07986934390929397865569047930, 6.43003979665691459292447923499, 6.56005756241627182633794128869, 7.67697086850144277688353697590, 8.270468870207606725632412141162, 9.01646523394236033972488365854, 9.75979695905120078040210364141, 11.044580223392450942866399342808, 11.26257892247876603024743772674, 12.48621875455017614174189390623, 12.95269230164990600836158138288, 13.611102998511664096605851228150, 14.24261061907629246955855924250, 15.20404898278925211152097652294, 15.7560071395504304330777089146, 16.63359892907629750260053085473, 17.70966390869185024557780995846, 18.038765912571643076817426303860, 18.653523211762498488888821390019, 19.61465955489108890338020594201

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