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Properties

Label 1-55e2-3025.1311-r1-0-0
Degree $1$
Conductor $3025$
Sign $0.999 + 0.00675i$
Analytic cond. $325.081$
Root an. cond. $325.081$
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  + (0.998 − 0.0570i)2-s + (−0.809 + 0.587i)3-s + (0.993 − 0.113i)4-s + (−0.774 + 0.633i)6-s + (−0.516 + 0.856i)7-s + (0.985 − 0.170i)8-s + (0.309 − 0.951i)9-s + (−0.736 + 0.676i)12-s + (−0.993 + 0.113i)13-s + (−0.466 + 0.884i)14-s + (0.974 − 0.226i)16-s + (0.0285 − 0.999i)17-s + (0.254 − 0.967i)18-s + (0.0285 + 0.999i)19-s + (−0.0855 − 0.996i)21-s + ⋯
L(s)  = 1  + (0.998 − 0.0570i)2-s + (−0.809 + 0.587i)3-s + (0.993 − 0.113i)4-s + (−0.774 + 0.633i)6-s + (−0.516 + 0.856i)7-s + (0.985 − 0.170i)8-s + (0.309 − 0.951i)9-s + (−0.736 + 0.676i)12-s + (−0.993 + 0.113i)13-s + (−0.466 + 0.884i)14-s + (0.974 − 0.226i)16-s + (0.0285 − 0.999i)17-s + (0.254 − 0.967i)18-s + (0.0285 + 0.999i)19-s + (−0.0855 − 0.996i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00675i$
Analytic conductor: \(325.081\)
Root analytic conductor: \(325.081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (1:\ ),\ 0.999 + 0.00675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.767146308 + 0.009339887095i\)
\(L(\frac12)\) \(\approx\) \(2.767146308 + 0.009339887095i\)
\(L(1)\) \(\approx\) \(1.414405864 + 0.2276027645i\)
\(L(1)\) \(\approx\) \(1.414405864 + 0.2276027645i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.998 - 0.0570i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.516 + 0.856i)T \)
13 \( 1 + (-0.993 + 0.113i)T \)
17 \( 1 + (0.0285 - 0.999i)T \)
19 \( 1 + (0.0285 + 0.999i)T \)
23 \( 1 + (0.516 + 0.856i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (0.198 + 0.980i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.362 + 0.931i)T \)
53 \( 1 + (0.516 - 0.856i)T \)
59 \( 1 + (-0.362 + 0.931i)T \)
61 \( 1 + (-0.774 - 0.633i)T \)
67 \( 1 + (-0.998 + 0.0570i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.897 + 0.441i)T \)
83 \( 1 + (0.921 - 0.389i)T \)
89 \( 1 + (-0.0285 + 0.999i)T \)
97 \( 1 + (-0.985 + 0.170i)T \)
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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

−19.11802165000378660873012659393, −17.99689265925736523827753490037, −17.28100340512340788317928057290, −16.73298717467485788018962748953, −16.20647823670809052794802414390, −15.33447597397424161441776435365, −14.5341741067558747864663428445, −13.847870452397468513274060499214, −13.04546568661082479761109918103, −12.69272201694585182958858249596, −12.03624734087996023390629704959, −11.17191915572291524559584994648, −10.530428225964378004872689273010, −10.07052149934085234593799298028, −8.67685253455928670702622577222, −7.647330168925940874233584698717, −7.051209218606898109739823047275, −6.5617727815842989878935968902, −5.83160795493066026946450569476, −4.78102368942505316356156618306, −4.55000171500414937306204929313, −3.32637813003121861602003373708, −2.58143287528147715165905818577, −1.551619481995923538623777092950, −0.67566363181072179093421829912, 0.45029944065987224409362659704, 1.72199111842535535299020306357, 2.73980707332546656426297657173, 3.369906226248704712462999013037, 4.35261146611796279507596239483, 5.00873448651949661341907783127, 5.64415534161203079242125730789, 6.26339672597923517111947879573, 7.0223538382624653972374335215, 7.853320255151779883183455545102, 9.13717178363548220233210652043, 9.8468440031173830755637253484, 10.33030914691177761200937993008, 11.45029614891515122528000372231, 11.901882279239484112423363423902, 12.27728103231041248019556679591, 13.19579257796588619378648901946, 13.934793681284239659510582371213, 14.92068802218515630723510859941, 15.30701171380488420553829937147, 15.93343337864796728773251924700, 16.677406075806392327935086884866, 17.15436265319582771533192903914, 18.18552250738268422753674171212, 18.98720958344364030970475020882

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