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Properties

Label 1-1375-1375.1172-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.677 - 0.735i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.684 − 0.728i)2-s + (0.998 − 0.0627i)3-s + (−0.0627 − 0.998i)4-s + (0.637 − 0.770i)6-s − i·7-s + (−0.770 − 0.637i)8-s + (0.992 − 0.125i)9-s + (−0.125 − 0.992i)12-s + (−0.125 − 0.992i)13-s + (−0.728 − 0.684i)14-s + (−0.992 + 0.125i)16-s + (0.844 − 0.535i)17-s + (0.587 − 0.809i)18-s + (0.535 + 0.844i)19-s + (−0.0627 − 0.998i)21-s + ⋯
L(s)  = 1  + (0.684 − 0.728i)2-s + (0.998 − 0.0627i)3-s + (−0.0627 − 0.998i)4-s + (0.637 − 0.770i)6-s − i·7-s + (−0.770 − 0.637i)8-s + (0.992 − 0.125i)9-s + (−0.125 − 0.992i)12-s + (−0.125 − 0.992i)13-s + (−0.728 − 0.684i)14-s + (−0.992 + 0.125i)16-s + (0.844 − 0.535i)17-s + (0.587 − 0.809i)18-s + (0.535 + 0.844i)19-s + (−0.0627 − 0.998i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.331243949 - 3.037000385i\)
\(L(\frac12)\) \(\approx\) \(1.331243949 - 3.037000385i\)
\(L(1)\) \(\approx\) \(1.602028517 - 1.350142329i\)
\(L(1)\) \(\approx\) \(1.602028517 - 1.350142329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.684 - 0.728i)T \)
3 \( 1 + (0.998 - 0.0627i)T \)
7 \( 1 - iT \)
13 \( 1 + (-0.125 - 0.992i)T \)
17 \( 1 + (0.844 - 0.535i)T \)
19 \( 1 + (0.535 + 0.844i)T \)
23 \( 1 + (0.904 + 0.425i)T \)
29 \( 1 + (-0.929 + 0.368i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (0.904 - 0.425i)T \)
41 \( 1 + (-0.728 + 0.684i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.248 + 0.968i)T \)
53 \( 1 + (-0.770 + 0.637i)T \)
59 \( 1 + (0.187 - 0.982i)T \)
61 \( 1 + (-0.876 + 0.481i)T \)
67 \( 1 + (-0.770 - 0.637i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (-0.904 - 0.425i)T \)
79 \( 1 + (-0.929 + 0.368i)T \)
83 \( 1 + (0.368 - 0.929i)T \)
89 \( 1 + (0.992 + 0.125i)T \)
97 \( 1 + (-0.248 + 0.968i)T \)
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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

−21.23009724857753710307538860366, −20.67530074971163624620906638633, −19.57991093982853372106955017417, −18.82774134700706207433502380456, −18.218931878010069737687025087298, −17.10407619031377411082790237534, −16.351598457479134413182168963431, −15.57939234397286500919348813812, −14.900399010199151548765586346265, −14.4989950276767966252007015400, −13.514446123924020957806260658364, −12.997139808698657622740481212549, −12.05999679529659761717807905895, −11.435563894198523927645076577584, −9.98000489156023413530167294048, −9.10377341322201158463501095692, −8.61233873757944405883203535515, −7.75096076420858717661006497381, −6.94739636930905898880733438477, −6.13020798804664018681036411059, −5.06991433179388918348281773804, −4.39726310742763832840493257269, −3.30542961394606215274979270013, −2.71120329680187434850237950364, −1.688485652055456252976899323239, 0.92197845859535272400088204750, 1.66436975611505121828037919830, 3.076118959293461680487472287293, 3.26417016931409045839092701227, 4.32192580423381970765570928346, 5.13757537025159091029851294735, 6.19556465458276055840656804372, 7.36667492450659432917991481568, 7.835217814752105933845078328518, 9.09319240680784631143368893232, 9.90176905169227913010305764629, 10.344384153076626662931795546259, 11.32324626860965953995771679393, 12.29393534978984236968053226746, 13.08826935780880785278785540320, 13.56772228094379602553431609766, 14.37706256977377738094874904351, 14.88157069901183527090820473, 15.7468035396536903259860804862, 16.64987008171413327645944173077, 17.77327863796420524617668246833, 18.67611491627039741096548328375, 19.19634309078604332388052594988, 20.1880794159402125792085738118, 20.41705094856799581440573158450

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