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Properties

Label 1-3520-3520.2309-r1-0-0
Degree $1$
Conductor $3520$
Sign $-0.995 - 0.0980i$
Analytic cond. $378.276$
Root an. cond. $378.276$
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)7-s + (−0.707 + 0.707i)9-s + (0.923 − 0.382i)13-s − i·17-s + (0.923 − 0.382i)19-s + (−0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s − 31-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)7-s + (−0.707 + 0.707i)9-s + (0.923 − 0.382i)13-s − i·17-s + (0.923 − 0.382i)19-s + (−0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s − 31-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.995 - 0.0980i$
Analytic conductor: \(378.276\)
Root analytic conductor: \(378.276\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (1:\ ),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05858800298 - 1.192586197i\)
\(L(\frac12)\) \(\approx\) \(0.05858800298 - 1.192586197i\)
\(L(1)\) \(\approx\) \(0.7686391804 - 0.3865384803i\)
\(L(1)\) \(\approx\) \(0.7686391804 - 0.3865384803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
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 \( 1 - iT \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
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 \( 1 + T \)
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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

−18.752154251236194117627303080927, −18.25523801015542436630431864369, −17.47896896867119764918244860657, −16.560297806549575111774158270293, −16.19722055899266635745355012886, −15.59756230248936937505150613330, −14.889584603917783034227840688536, −14.23967718443644042646025488211, −13.31502864313393505429659980062, −12.574380177246201343189092546354, −11.794578931945372529395820190917, −11.32689556649728361105716117665, −10.25547246966752802519565734544, −10.022929607865741876021803257900, −8.93772826528673928598452353393, −8.698631119649301925559179177854, −7.61666513016259343298042874119, −6.45365807980467268752269531384, −5.961425007338482888216566918039, −5.44774898052366560866296171181, −4.28516098256566351600580371876, −3.79436565628195990509571587525, −2.99315816377036197858255794141, −2.03490262643029533283006723147, −0.82882579063508992825326701609, 0.26878083925796849291735445742, 0.95196189995486138899447680088, 1.75969263240298564075043201029, 2.96721475475449939227255584217, 3.429637864446141138219334902942, 4.599543012737247217730850629950, 5.48676792317895468435883393969, 6.10873738451165030382385671298, 6.95130022458055038506123237518, 7.39579942838728018513847317409, 8.15316160236333890339289734351, 9.082572159798482573257266931519, 9.822625551108766085409904812917, 10.6869938665176383520341143490, 11.324922401512089498148209830967, 11.96726893771680187839299982380, 12.79856470567205309437510964592, 13.432819648860991615967663658195, 13.781535957348872152047588032449, 14.58844144986845444329764061297, 15.8139825974449136934383277619, 16.17802993119323302805188379973, 16.8424402052222565774429830334, 17.78668830475356273770486920049, 18.138731178617509689295492041536

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