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Properties

Label 1-1375-1375.217-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  + (−0.684 − 0.728i)2-s + (−0.248 − 0.968i)3-s + (−0.0627 + 0.998i)4-s + (−0.535 + 0.844i)6-s + (0.587 + 0.809i)7-s + (0.770 − 0.637i)8-s + (−0.876 + 0.481i)9-s + (0.982 − 0.187i)12-s + (0.481 + 0.876i)13-s + (0.187 − 0.982i)14-s + (−0.992 − 0.125i)16-s + (0.248 − 0.968i)17-s + (0.951 + 0.309i)18-s + (0.535 − 0.844i)19-s + (0.637 − 0.770i)21-s + ⋯
L(s)  = 1  + (−0.684 − 0.728i)2-s + (−0.248 − 0.968i)3-s + (−0.0627 + 0.998i)4-s + (−0.535 + 0.844i)6-s + (0.587 + 0.809i)7-s + (0.770 − 0.637i)8-s + (−0.876 + 0.481i)9-s + (0.982 − 0.187i)12-s + (0.481 + 0.876i)13-s + (0.187 − 0.982i)14-s + (−0.992 − 0.125i)16-s + (0.248 − 0.968i)17-s + (0.951 + 0.309i)18-s + (0.535 − 0.844i)19-s + (0.637 − 0.770i)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: $\chi_{1375} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.742 - 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9184349114 - 0.3534374151i\)
\(L(\frac12)\) \(\approx\) \(0.9184349114 - 0.3534374151i\)
\(L(1)\) \(\approx\) \(0.7004470290 - 0.3020604392i\)
\(L(1)\) \(\approx\) \(0.7004470290 - 0.3020604392i\)

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 \( 1 + (-0.248 - 0.968i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.481 + 0.876i)T \)
17 \( 1 + (0.248 - 0.968i)T \)
19 \( 1 + (0.535 - 0.844i)T \)
23 \( 1 + (0.125 + 0.992i)T \)
29 \( 1 + (-0.929 - 0.368i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.481 + 0.876i)T \)
41 \( 1 + (-0.876 + 0.481i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.844 + 0.535i)T \)
53 \( 1 + (0.844 - 0.535i)T \)
59 \( 1 + (0.425 - 0.904i)T \)
61 \( 1 + (0.425 + 0.904i)T \)
67 \( 1 + (-0.248 + 0.968i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (0.684 + 0.728i)T \)
79 \( 1 + (-0.637 + 0.770i)T \)
83 \( 1 + (0.770 - 0.637i)T \)
89 \( 1 + (0.187 - 0.982i)T \)
97 \( 1 + (0.998 + 0.0627i)T \)
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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

−20.70066095965776179929278875602, −20.25287511938363577275883588807, −19.5008418008980458083821803404, −18.270796455107517975343309178652, −17.87059053022577638365679677368, −16.86325995581126640413385786080, −16.59179730060462020468768919335, −15.76248989196479008411749717025, −14.78713535974947274643506843041, −14.5708441187926309510695457901, −13.56911133796331592525060264837, −12.43024537367856895608866045836, −11.1786296722791773535354467989, −10.6360170079563787470155559359, −10.18233937310800283667824453801, −9.2221966946809264998209589254, −8.3512278197200796598477712475, −7.7971370638247931339375806487, −6.72639666443282830470421049432, −5.756503817465860611951488107388, −5.202392307833616965388408841766, −4.18763081866415794931661269843, −3.400549767551921218356133584805, −1.7813606542540939900978061285, −0.668141660111687626581901266107, 0.89406058658956418105386353276, 1.82857888287531023147762182936, 2.48752226404825215227386105926, 3.49854880539791847683350109982, 4.80738687969628997989733253950, 5.66654770268873726538178866298, 6.8249058337480628819979919934, 7.48819051377153289699744637486, 8.28314662394911480488547683063, 9.07759130133563712215964765452, 9.6930531532322096247218085633, 11.13422518467654158687974572047, 11.5408200580487642383801598654, 11.904682478203669924955589818460, 13.07128386794612715573291101298, 13.510330731256820494619486000815, 14.46721779391237913021507675351, 15.62250436853508814995098316361, 16.48062499617362102474150007767, 17.269943903610424903902691534143, 17.95243719481927822274092380375, 18.62894834233629249209502144512, 18.914983104859242495467650476162, 19.92796188773753749217199182025, 20.59185235271519691871662883409

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