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Properties

Label 1-2496-2496.2291-r0-0-0
Degree $1$
Conductor $2496$
Sign $-0.985 - 0.169i$
Analytic cond. $11.5913$
Root an. cond. $11.5913$
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)5-s + (−0.965 − 0.258i)7-s + (0.991 + 0.130i)11-s + (−0.866 + 0.5i)17-s + (−0.130 − 0.991i)19-s + (−0.965 + 0.258i)23-s + (0.707 − 0.707i)25-s + (0.608 + 0.793i)29-s + 31-s + (0.991 − 0.130i)35-s + (0.130 − 0.991i)37-s + (0.965 − 0.258i)41-s + (−0.608 + 0.793i)43-s − i·47-s + (0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)5-s + (−0.965 − 0.258i)7-s + (0.991 + 0.130i)11-s + (−0.866 + 0.5i)17-s + (−0.130 − 0.991i)19-s + (−0.965 + 0.258i)23-s + (0.707 − 0.707i)25-s + (0.608 + 0.793i)29-s + 31-s + (0.991 − 0.130i)35-s + (0.130 − 0.991i)37-s + (0.965 − 0.258i)41-s + (−0.608 + 0.793i)43-s − i·47-s + (0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (2291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ -0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001084274154 + 0.01272268913i\)
\(L(\frac12)\) \(\approx\) \(0.001084274154 + 0.01272268913i\)
\(L(1)\) \(\approx\) \(0.7029746469 + 0.03730318288i\)
\(L(1)\) \(\approx\) \(0.7029746469 + 0.03730318288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (0.991 + 0.130i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.130 - 0.991i)T \)
23 \( 1 + (-0.965 + 0.258i)T \)
29 \( 1 + (0.608 + 0.793i)T \)
31 \( 1 + T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.965 - 0.258i)T \)
43 \( 1 + (-0.608 + 0.793i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (-0.793 - 0.608i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (-0.608 - 0.793i)T \)
71 \( 1 + (0.965 + 0.258i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.138869475888079465178589387312, −18.58936003421084950353873671785, −17.62856821722063392068174524679, −16.69646399856776381677253527302, −16.322126204748947832561753648703, −15.52869906155138367822059192624, −15.022629312259231372148594625864, −13.97078304511426315523451921107, −13.37446997315366596594889132898, −12.37123373516391224068700686458, −11.97986542061830458870224836091, −11.364420660200154180906087201278, −10.2279298729296918459256841068, −9.640282644879950547721396636592, −8.70649472144082982619035786520, −8.26665649320496376195510022230, −7.22229498205730968300890688608, −6.48378153254019652821426644430, −5.87055698408728945980026332965, −4.634231082836763574623964271516, −4.04337686757811790914394710101, −3.30383917198217252267825438986, −2.34466971561842763157339052417, −1.127613946561312116983287140, −0.00483701470142872912829705187, 1.19984543455063091721965734884, 2.51669232036520757834212392281, 3.25712904382736115885009371312, 4.13457328558355404431126109094, 4.55311984818810002222629818204, 6.05445303693757647010210714138, 6.56991098806308944114768568419, 7.219015029866310500959199256, 8.069307137713364217216352342188, 8.98326594973108633713294843448, 9.541305094550846668522411929977, 10.60001994442120517328992243870, 11.06746501765310651162366525457, 12.02371822552170717336611651739, 12.490372717770494689767732720883, 13.40153547316195116019491602025, 14.14532673847424041731549681273, 14.89881835057358216216838207438, 15.75221362872116155478374411001, 16.013285652410574330890176378859, 17.01744886907761264813135863284, 17.66214358206242725102600770354, 18.489753874144668955041047209129, 19.40422645200077042083725863778, 19.8073023292318338991664487753

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