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Properties

Label 1-5600-5600.4381-r1-0-0
Degree $1$
Conductor $5600$
Sign $-0.943 - 0.331i$
Analytic cond. $601.803$
Root an. cond. $601.803$
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  + (−0.891 + 0.453i)3-s + (0.587 − 0.809i)9-s + (−0.156 + 0.987i)11-s + (−0.156 − 0.987i)13-s + (0.309 + 0.951i)17-s + (−0.453 + 0.891i)19-s + (−0.587 − 0.809i)23-s + (−0.156 + 0.987i)27-s + (0.891 − 0.453i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (0.987 − 0.156i)37-s + (0.587 + 0.809i)39-s + (−0.587 + 0.809i)41-s + (0.707 − 0.707i)43-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)3-s + (0.587 − 0.809i)9-s + (−0.156 + 0.987i)11-s + (−0.156 − 0.987i)13-s + (0.309 + 0.951i)17-s + (−0.453 + 0.891i)19-s + (−0.587 − 0.809i)23-s + (−0.156 + 0.987i)27-s + (0.891 − 0.453i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (0.987 − 0.156i)37-s + (0.587 + 0.809i)39-s + (−0.587 + 0.809i)41-s + (0.707 − 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(601.803\)
Root analytic conductor: \(601.803\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (4381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (1:\ ),\ -0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01732738418 + 0.1016363751i\)
\(L(\frac12)\) \(\approx\) \(0.01732738418 + 0.1016363751i\)
\(L(1)\) \(\approx\) \(0.7222500139 + 0.1147321808i\)
\(L(1)\) \(\approx\) \(0.7222500139 + 0.1147321808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.891 + 0.453i)T \)
11 \( 1 + (-0.156 + 0.987i)T \)
13 \( 1 + (-0.156 - 0.987i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.453 + 0.891i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.987 - 0.156i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (-0.987 + 0.156i)T \)
61 \( 1 + (0.987 + 0.156i)T \)
67 \( 1 + (-0.453 + 0.891i)T \)
71 \( 1 + (-0.951 - 0.309i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.453 + 0.891i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−17.298211330451952082104873844267, −16.69680980452429000000251742558, −15.96938778521933185423942596740, −15.76299730014718113890777221141, −14.486809763393164991591000557548, −13.85123877519042108996322962539, −13.453531989698072208266344282086, −12.50805455571343005428461498273, −12.00663864404832969895615250056, −11.24773797132681890265941169762, −10.941126967528349252603647248072, −10.03445466053225929285808549515, −9.2450674108623894957093055611, −8.586412317370173313899216802868, −7.60683764189318260412684887195, −7.14789934723681299257879060619, −6.29594825125786762838049788622, −5.8604634869942729442602060053, −4.87927611802852061924506559203, −4.509878828501098342938688726915, −3.34221317194587350952433221642, −2.56465912867717491474709169439, −1.61194537449612149787494261022, −0.84779349496509239228582557287, −0.022683222259660901182702290009, 0.83664398764565578761976408683, 1.8144830777153412734842231478, 2.67280715161671070618195843015, 3.81183496979658388719372686385, 4.24786231573153325966071212435, 5.04875626276008099826126871921, 5.829693755488245853261661613988, 6.25755839626199452734336603389, 7.15399928909691196400506776274, 7.94074517128982651697335332330, 8.54414727909419703206272268677, 9.701275294661188427080097834914, 10.137068948819031921841165704113, 10.50909729008703758731403635834, 11.41040226445088392665591715501, 12.1252040980467589389936798910, 12.67852212646416254642654397685, 13.08385486166456299158889935256, 14.3361422404189496239245168995, 14.903406597111366385489960190575, 15.40155841585475334211272661684, 16.078636969030798350342185102743, 16.99976050332222861215805782681, 17.10811823323242147498355065567, 18.121661351493356481170986254989

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