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Properties

Label 1-5600-5600.1139-r0-0-0
Degree $1$
Conductor $5600$
Sign $-0.270 + 0.962i$
Analytic cond. $26.0062$
Root an. cond. $26.0062$
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  + (0.933 + 0.358i)3-s + (0.743 + 0.669i)9-s + (0.998 + 0.0523i)11-s + (−0.453 + 0.891i)13-s + (0.104 + 0.994i)17-s + (0.358 + 0.933i)19-s + (−0.207 − 0.978i)23-s + (0.453 + 0.891i)27-s + (0.156 + 0.987i)29-s + (−0.104 − 0.994i)31-s + (0.913 + 0.406i)33-s + (0.0523 + 0.998i)37-s + (−0.743 + 0.669i)39-s + (0.951 − 0.309i)41-s + (−0.707 + 0.707i)43-s + ⋯
L(s)  = 1  + (0.933 + 0.358i)3-s + (0.743 + 0.669i)9-s + (0.998 + 0.0523i)11-s + (−0.453 + 0.891i)13-s + (0.104 + 0.994i)17-s + (0.358 + 0.933i)19-s + (−0.207 − 0.978i)23-s + (0.453 + 0.891i)27-s + (0.156 + 0.987i)29-s + (−0.104 − 0.994i)31-s + (0.913 + 0.406i)33-s + (0.0523 + 0.998i)37-s + (−0.743 + 0.669i)39-s + (0.951 − 0.309i)41-s + (−0.707 + 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.270 + 0.962i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ -0.270 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.588943838 + 2.096687713i\)
\(L(\frac12)\) \(\approx\) \(1.588943838 + 2.096687713i\)
\(L(1)\) \(\approx\) \(1.438004032 + 0.5169755867i\)
\(L(1)\) \(\approx\) \(1.438004032 + 0.5169755867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.933 + 0.358i)T \)
11 \( 1 + (0.998 + 0.0523i)T \)
13 \( 1 + (-0.453 + 0.891i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.358 + 0.933i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.0523 + 0.998i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.358 - 0.933i)T \)
59 \( 1 + (-0.838 + 0.544i)T \)
61 \( 1 + (-0.838 - 0.544i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.987 + 0.156i)T \)
89 \( 1 + (-0.207 - 0.978i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−17.82828859519890800089245352238, −17.125810990488135196802873535772, −16.19429088238041289453167264933, −15.5234538858728093756986477651, −15.006599482032439248071578198653, −14.292717355234423941717547370475, −13.64951468391405444665447128589, −13.284708859830270660334723382446, −12.19096258778273715246619885603, −11.980549496209161416265515365928, −10.99584736795645910095241394770, −10.140205422717445525825966763985, −9.29257211845829427033936426419, −9.15831951403480803046358870353, −8.14541459514105396269153759771, −7.46743775556874859206838392077, −7.033058719775432953540662719536, −6.17274811770810087063438040573, −5.28917199364626967292528544774, −4.48115010011790308491091404634, −3.62679317144983275530964555822, −3.00985192176277125145123127284, −2.31576692942931302600336022457, −1.3886306567941986746476323656, −0.568948362151290711474997114875, 1.328332459549138344378768769521, 1.816800070388452965068035713533, 2.75776750020776922502522141507, 3.54307985519400845763842304992, 4.23611256009148575591006840281, 4.673502185781151654853236695845, 5.849452606864593643879082627699, 6.54667610409704130590184491445, 7.27841105998439678683595003348, 8.08733225488172256349200659947, 8.60822656960180810429394933827, 9.3921745594726982570633934419, 9.8229512896300127332871676309, 10.5816479795895552496605654734, 11.35155841870672496806446620249, 12.219657263012553990695940525111, 12.69229486904358673517401847188, 13.60093337680445065145931897212, 14.29197036881627780922006397276, 14.63313173531536057267333116291, 15.170925276174789607726204265491, 16.19535829533538262139422440356, 16.604960026542497574658789958427, 17.16990934357423983695945198802, 18.23768038073504867791121084642

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