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Properties

Label 1-1400-1400.13-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.728 - 0.684i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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  + (−0.587 − 0.809i)3-s + (−0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.951 + 0.309i)23-s + (0.951 − 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.587 + 0.809i)33-s + (0.951 + 0.309i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + (−0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.951 + 0.309i)23-s + (0.951 − 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.587 + 0.809i)33-s + (0.951 + 0.309i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.172501280 - 0.4642260973i\)
\(L(\frac12)\) \(\approx\) \(1.172501280 - 0.4642260973i\)
\(L(1)\) \(\approx\) \(0.9099137198 - 0.2311380968i\)
\(L(1)\) \(\approx\) \(0.9099137198 - 0.2311380968i\)

Euler product

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

−20.76184350079441533570870772241, −20.47351359221256603504206808226, −19.475981324474716923818918513739, −18.37478167320297583415640753663, −17.82344819548150764890449754541, −17.12059331630940961205585951846, −16.26312710892298410346279149713, −15.618065215899249682590671477597, −15.026047808444706542234119028620, −14.17291626532020116721104915539, −13.11729138400261614853374034254, −12.401337287950523208768941879949, −11.50860506115185407481205105021, −10.89339887749923307172414202384, −9.92622395763638707989260112140, −9.59214383132912286049023674879, −8.40099516621801423364062493625, −7.64068346366749752965582764771, −6.44697237949030750432547585684, −5.81574078843911949963549168783, −4.9408881894472163187789009754, −4.095060167200525196353935577809, −3.35665704995463109004442340831, −2.12013304484473099927866472096, −0.804845646736929244243929889, 0.798433891002762721723304876507, 1.624892886212616881966039005432, 2.84309407804847892827360796262, 3.7145610056256246835759042944, 5.02937999267510562336041621815, 5.755002578812436332066959963418, 6.37439257733772362259786023015, 7.39869603458239177020445727096, 8.04700704282115278563122154127, 8.86992589127952794546443477494, 10.00270315956628961139170197526, 10.81484851924653515841839323, 11.68052438428827716293653232557, 12.02124871859979307005473801975, 13.230585906078546293339492766030, 13.672716142444432018769087113383, 14.334665446560422429743063153632, 15.6204818684405414117237581513, 16.464077591135599250073771731619, 16.66925676340724560724225310893, 18.091796133280184229802322670088, 18.271092432558735481145069978349, 18.97965122931784399360674381727, 19.87767852700836385209247412087, 20.678265480036899009469718537696

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