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Properties

Label 1-2600-2600.37-r0-0-0
Degree $1$
Conductor $2600$
Sign $-0.390 - 0.920i$
Analytic cond. $12.0743$
Root an. cond. $12.0743$
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  + (0.994 − 0.104i)3-s + (−0.5 − 0.866i)7-s + (0.978 − 0.207i)9-s + (0.207 − 0.978i)11-s + (0.994 + 0.104i)17-s + (−0.994 − 0.104i)19-s + (−0.587 − 0.809i)21-s + (0.207 − 0.978i)23-s + (0.951 − 0.309i)27-s + (−0.104 − 0.994i)29-s + (0.587 − 0.809i)31-s + (0.104 − 0.994i)33-s + (−0.669 − 0.743i)37-s + (−0.743 + 0.669i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)3-s + (−0.5 − 0.866i)7-s + (0.978 − 0.207i)9-s + (0.207 − 0.978i)11-s + (0.994 + 0.104i)17-s + (−0.994 − 0.104i)19-s + (−0.587 − 0.809i)21-s + (0.207 − 0.978i)23-s + (0.951 − 0.309i)27-s + (−0.104 − 0.994i)29-s + (0.587 − 0.809i)31-s + (0.104 − 0.994i)33-s + (−0.669 − 0.743i)37-s + (−0.743 + 0.669i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $-0.390 - 0.920i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ -0.390 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119345970 - 1.690749583i\)
\(L(\frac12)\) \(\approx\) \(1.119345970 - 1.690749583i\)
\(L(1)\) \(\approx\) \(1.304407234 - 0.4691312189i\)
\(L(1)\) \(\approx\) \(1.304407234 - 0.4691312189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.994 - 0.104i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.994 + 0.104i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (0.207 - 0.978i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
41 \( 1 + (-0.743 + 0.669i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.207 - 0.978i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−19.52422984430330627849043497898, −18.980371502337855386204875550708, −18.417070934249619167310069224121, −17.517267511915047350894912611305, −16.69192300173077499032403693521, −15.8084195823689082250321034806, −15.24812627095704014424646804349, −14.751194304424662460647757153254, −13.96048442654048038577042393592, −13.18245077966642477203135346382, −12.37369628212077767110940282563, −12.067326917106473639190366925551, −10.76083556420889237284940102346, −9.92455626535206437985698484639, −9.50451991331456596724546448318, −8.648865012706960057010503701636, −8.11678056438876752812466316778, −7.07930283299893634320189144206, −6.58925919411004150753229941854, −5.36021623763269077927715449783, −4.74115383099726989050510609452, −3.561701137901747511177689121444, −3.15239763815278453557020014783, −2.07084630875155330035369852787, −1.501960866605730331349863467418, 0.53285876322685409175623684490, 1.51251233885233757571225000481, 2.60206253564213704405722145464, 3.34528112251735130774038773518, 3.98044176980071881019165130678, 4.77914180444712544095095673704, 6.14107929573468786020564813780, 6.598360719243755562464934002078, 7.60491431376970819376926679994, 8.184411727210972223174519918306, 8.86583841809044269081409023193, 9.79320809213257521231903537527, 10.27686392133411133523174085767, 11.12212988561567822168578048326, 12.10457187591667245458537520185, 13.00464217075387443139344402248, 13.40715958265754848867001519424, 14.18013101974290480575683283450, 14.7312654457638636164371018820, 15.50573553854780066668526169188, 16.51464258924457449473409181877, 16.7274292803276754291562610899, 17.7914153377130156858425601147, 18.781685215493369525865075864621, 19.22566970665528142288537076091

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