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Properties

Label 1-6040-6040.707-r0-0-0
Degree $1$
Conductor $6040$
Sign $-0.557 - 0.830i$
Analytic cond. $28.0496$
Root an. cond. $28.0496$
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  + (0.368 + 0.929i)3-s + (−0.999 + 0.0209i)7-s + (−0.728 + 0.684i)9-s + (0.146 + 0.989i)11-s + (−0.886 + 0.463i)13-s + (0.518 − 0.855i)17-s + (0.809 + 0.587i)19-s + (−0.387 − 0.921i)21-s + (−0.994 − 0.104i)23-s + (−0.904 − 0.425i)27-s + (−0.637 + 0.770i)29-s + (0.756 − 0.653i)31-s + (−0.866 + 0.5i)33-s + (−0.0418 + 0.999i)37-s + (−0.756 − 0.653i)39-s + ⋯
L(s)  = 1  + (0.368 + 0.929i)3-s + (−0.999 + 0.0209i)7-s + (−0.728 + 0.684i)9-s + (0.146 + 0.989i)11-s + (−0.886 + 0.463i)13-s + (0.518 − 0.855i)17-s + (0.809 + 0.587i)19-s + (−0.387 − 0.921i)21-s + (−0.994 − 0.104i)23-s + (−0.904 − 0.425i)27-s + (−0.637 + 0.770i)29-s + (0.756 − 0.653i)31-s + (−0.866 + 0.5i)33-s + (−0.0418 + 0.999i)37-s + (−0.756 − 0.653i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
Sign: $-0.557 - 0.830i$
Analytic conductor: \(28.0496\)
Root analytic conductor: \(28.0496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6040} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6040,\ (0:\ ),\ -0.557 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2864308704 + 0.5371913972i\)
\(L(\frac12)\) \(\approx\) \(-0.2864308704 + 0.5371913972i\)
\(L(1)\) \(\approx\) \(0.7607036427 + 0.4542011126i\)
\(L(1)\) \(\approx\) \(0.7607036427 + 0.4542011126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
151 \( 1 \)
good3 \( 1 + (0.368 + 0.929i)T \)
7 \( 1 + (-0.999 + 0.0209i)T \)
11 \( 1 + (0.146 + 0.989i)T \)
13 \( 1 + (-0.886 + 0.463i)T \)
17 \( 1 + (0.518 - 0.855i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (-0.637 + 0.770i)T \)
31 \( 1 + (0.756 - 0.653i)T \)
37 \( 1 + (-0.0418 + 0.999i)T \)
41 \( 1 + (0.0627 + 0.998i)T \)
43 \( 1 + (0.999 + 0.0209i)T \)
47 \( 1 + (-0.444 + 0.895i)T \)
53 \( 1 + (0.248 + 0.968i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.783 + 0.621i)T \)
67 \( 1 + (-0.684 + 0.728i)T \)
71 \( 1 + (0.855 - 0.518i)T \)
73 \( 1 + (0.481 - 0.876i)T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (-0.125 + 0.992i)T \)
89 \( 1 + (-0.604 + 0.796i)T \)
97 \( 1 + (0.973 + 0.228i)T \)
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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

−17.290482797506422445827173657792, −16.75750687365305679646295879189, −15.911179394190966230581206520639, −15.36473189700432483250629985301, −14.43682575562999071149571517355, −13.95706947079192790561388439260, −13.31939696848781929792985506322, −12.71412021267144241486900777741, −12.1246924313926527357449263095, −11.57522666372776669216709869641, −10.577534772649086423549261224541, −9.91458727347606564765185296449, −9.18563790135658789051287335630, −8.573972064185961485781048069050, −7.735290695652577587736633180374, −7.320446842054973944184955742153, −6.4197119330298514497298876021, −5.903853132593767656843834386565, −5.30057027589721162848915243483, −3.9432173834777306449088194754, −3.38809539360460321682773062107, −2.71029712221480699469603665175, −1.980685560856080389086307792219, −0.88958445703810335194295585353, −0.1659982777489762631019617802, 1.356554011799219017262943603834, 2.47066045266564871056584317281, 2.92154038901958539101673290748, 3.78469902103892591337888318705, 4.44366033258550217598399472183, 5.095794496745868518496207523234, 5.8861192274471544398016637506, 6.69508513566318144925156112837, 7.544292611678485884587740192022, 7.99958388115793495496418715305, 9.23392649566412252217420842480, 9.551790218609069080214737024403, 9.91316149770299932148136923075, 10.63848712499269281811680088358, 11.67679576135062691712742905503, 12.12299097066467250739497763009, 12.826437555271552592580982783, 13.83746654144615281400315563157, 14.15849180024479197827387319023, 14.98381799613074902906030219927, 15.47945079829443337867318945692, 16.29417303066087970095286393421, 16.5618732798393183834077838171, 17.28932142874481108080219831813, 18.18598818013975387758500613020

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