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Properties

Label 1-1375-1375.997-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.742 + 0.670i$
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  + (−0.125 + 0.992i)2-s + (−0.844 − 0.535i)3-s + (−0.968 − 0.248i)4-s + (0.637 − 0.770i)6-s + (−0.587 + 0.809i)7-s + (0.368 − 0.929i)8-s + (0.425 + 0.904i)9-s + (0.684 + 0.728i)12-s + (0.904 − 0.425i)13-s + (−0.728 − 0.684i)14-s + (0.876 + 0.481i)16-s + (0.844 − 0.535i)17-s + (−0.951 + 0.309i)18-s + (−0.637 + 0.770i)19-s + (0.929 − 0.368i)21-s + ⋯
L(s)  = 1  + (−0.125 + 0.992i)2-s + (−0.844 − 0.535i)3-s + (−0.968 − 0.248i)4-s + (0.637 − 0.770i)6-s + (−0.587 + 0.809i)7-s + (0.368 − 0.929i)8-s + (0.425 + 0.904i)9-s + (0.684 + 0.728i)12-s + (0.904 − 0.425i)13-s + (−0.728 − 0.684i)14-s + (0.876 + 0.481i)16-s + (0.844 − 0.535i)17-s + (−0.951 + 0.309i)18-s + (−0.637 + 0.770i)19-s + (0.929 − 0.368i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.742 + 0.670i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.742 + 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7439574873 + 0.2862940074i\)
\(L(\frac12)\) \(\approx\) \(0.7439574873 + 0.2862940074i\)
\(L(1)\) \(\approx\) \(0.6427032988 + 0.2279095066i\)
\(L(1)\) \(\approx\) \(0.6427032988 + 0.2279095066i\)

Euler product

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

−20.889334086198644792813332516922, −20.05314795354128297758283492148, −19.30020800784216439121290561150, −18.56937221390844008420790914923, −17.67571509737625116283846145317, −17.089488881626372633802437935069, −16.420664181049098100424880095171, −15.60925427119136304384256585546, −14.52716958434232204699721291130, −13.61061295834423804882345466198, −12.92850014616753922703854792449, −12.1977122397721109739352112391, −11.19310711458496518600381529345, −10.90898237641193673866375600413, −9.91748165427404764384716158985, −9.51957485137976387269165060969, −8.47647675597703364554671018387, −7.39905064064058143568559564211, −6.310392548236875349239529016975, −5.5608357641460977654243167121, −4.384990299424381023077173395421, −3.909540320970697174141348704570, −3.10236517119980094734044986063, −1.641915591472653735255700291239, −0.7021515653239592075013964657, 0.61657713557625841654209617374, 1.82678009408448920419769390197, 3.25150966542050275684596340860, 4.3637441941781382033654434354, 5.44791593262819077169438080492, 5.91003733113340722838585752971, 6.542951263335173242034648888658, 7.48002098450618690614658226393, 8.24762048063060780810314478391, 9.05136387673432196407127353168, 10.05487124680334568832593230955, 10.74600573664845334282251755416, 11.91068078100630059700574860904, 12.67016878289770769402480728302, 13.12998318658063431021300590578, 14.16520034760237045646321141184, 14.92941448947670460648171860350, 15.91689761023052852432434746779, 16.38637634182500026226716499663, 16.93838322425137086213298250114, 18.13228392705074247458424927776, 18.386909943413693896209612485099, 18.92678551679499307972514019400, 19.96256039158890101855925323027, 21.175811163317365041827229763629

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