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Properties

Label 1-1575-1575.1517-r1-0-0
Degree $1$
Conductor $1575$
Sign $-0.812 + 0.583i$
Analytic cond. $169.257$
Root an. cond. $169.257$
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)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)8-s + (−0.913 − 0.406i)11-s + (−0.994 − 0.104i)13-s + (−0.809 − 0.587i)16-s + (0.207 − 0.978i)17-s + (0.669 − 0.743i)19-s + (−0.207 − 0.978i)22-s + (0.994 − 0.104i)23-s + (−0.5 − 0.866i)26-s + (0.669 + 0.743i)29-s + (−0.309 − 0.951i)31-s − i·32-s + (0.913 − 0.406i)34-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)8-s + (−0.913 − 0.406i)11-s + (−0.994 − 0.104i)13-s + (−0.809 − 0.587i)16-s + (0.207 − 0.978i)17-s + (0.669 − 0.743i)19-s + (−0.207 − 0.978i)22-s + (0.994 − 0.104i)23-s + (−0.5 − 0.866i)26-s + (0.669 + 0.743i)29-s + (−0.309 − 0.951i)31-s − i·32-s + (0.913 − 0.406i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.812 + 0.583i$
Analytic conductor: \(169.257\)
Root analytic conductor: \(169.257\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (1:\ ),\ -0.812 + 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5009084160 + 1.557239510i\)
\(L(\frac12)\) \(\approx\) \(0.5009084160 + 1.557239510i\)
\(L(1)\) \(\approx\) \(1.027417680 + 0.5531216712i\)
\(L(1)\) \(\approx\) \(1.027417680 + 0.5531216712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.994 - 0.104i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.743 + 0.669i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−20.16795014331018299275330514623, −19.16553378319221761465897848714, −18.97786292183717602126780292500, −17.80997393507672259202981891449, −17.29389239566631211383448596804, −16.08376146618526806000827228385, −15.34137413010627877623173154551, −14.64593826447907283216114534596, −13.9402785949638470803016050499, −13.105094804238230306653822279538, −12.34278375147031989713894994266, −11.96033539850875712127515775292, −10.72166273873603930302776931037, −10.335828826482141838129620131405, −9.54105062546401132018617447806, −8.613114962658749605646848442727, −7.578355896100183754961821852088, −6.71060628839305351099461402656, −5.54063494669189859384896374640, −5.1022250666793827078237462217, −4.11227270737351214622624146881, −3.22323585203021632186240826436, −2.36454695204929674923186789564, −1.5229973141808817702009794624, −0.30863537952004007236846966442, 0.790313581996253724976317195436, 2.67788989426978972517943339961, 2.93471250945443553274825247404, 4.26717041360683558041439984511, 5.13981588849992309444370485655, 5.501107046340613280141635552004, 6.76989431394740462405855229027, 7.332695005030928292385970973485, 8.04235419821478528155805076882, 9.02082255240615130800091526236, 9.70492627304907320781562538122, 10.87266873021599797089507467201, 11.67277763369280837554541310109, 12.52782902392672259305219484432, 13.17718366555798824664371712799, 13.941576936127324948899178961102, 14.58666511976752719902965541159, 15.48326718562130132403243110321, 15.98198029330211289564432313995, 16.7738241684887934230959852484, 17.51681286402799277730027258158, 18.23008670246920973824588259892, 18.97174640073964246375337012434, 20.03937944771224220505033547658, 20.82402770988964241239067147487

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