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Properties

Label 1-1575-1575.1096-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.0911 - 0.995i$
Analytic cond. $7.31426$
Root an. cond. $7.31426$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)11-s + (−0.104 − 0.994i)13-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + (0.669 + 0.743i)19-s + (−0.978 − 0.207i)22-s + (−0.104 + 0.994i)23-s + (−0.5 + 0.866i)26-s + (0.669 − 0.743i)29-s + (0.309 − 0.951i)31-s + 32-s + (0.913 + 0.406i)34-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)11-s + (−0.104 − 0.994i)13-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + (0.669 + 0.743i)19-s + (−0.978 − 0.207i)22-s + (−0.104 + 0.994i)23-s + (−0.5 + 0.866i)26-s + (0.669 − 0.743i)29-s + (0.309 − 0.951i)31-s + 32-s + (0.913 + 0.406i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0911 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0911 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.0911 - 0.995i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1096, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.0911 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6452119604 - 0.7069828874i\)
\(L(\frac12)\) \(\approx\) \(0.6452119604 - 0.7069828874i\)
\(L(1)\) \(\approx\) \(0.7135859497 - 0.2638712501i\)
\(L(1)\) \(\approx\) \(0.7135859497 - 0.2638712501i\)

Euler product

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

−20.2636843946339576705416967299, −19.93177381525679221146808438249, −19.1908975693215927200358064662, −18.25258571276345223711871594405, −17.823409619083791699199763183374, −16.89593091780660379460079397650, −16.39674097886502123330234381657, −15.57225023041542434697035601914, −14.82385765015971072535747296825, −14.11768957092598438019277408608, −13.439390773761953180868434580212, −12.10501658141639464570709726618, −11.54939056681095743808622782822, −10.66265954220097467959277032186, −9.833607414742300302992135750813, −9.01798904416014145235494542120, −8.63553782350707586886356768887, −7.46268079127087698390236466710, −6.654081943316037631969900048273, −6.38833204623105207035743603519, −4.89988577770700964903826596744, −4.50467683437821557480535443642, −2.99221269746278326242177624224, −1.93187887485458927716577485734, −1.05414568404467819064175658604, 0.545166958014843654659160538425, 1.58405480777595863150125854970, 2.542448562426016496986279706757, 3.53937493379778053848158443902, 4.15670162325296015664980441346, 5.52777431191850443925593264559, 6.40216261102970768663409281566, 7.375573469857564039417644788351, 8.071669754904243247727494329664, 8.88031322560782683144019615415, 9.605507531313738861720072465867, 10.34408338534589464873469839056, 11.16955635501227315948928211023, 11.81960672557275983554089644898, 12.5450062164128858171856017038, 13.42615141395369073049590312149, 14.137339769428415799454370136522, 15.40828173070633096797744479390, 15.802803440176694905855239713364, 16.94963041283243308749254068833, 17.33342233214096413432660995498, 18.10174882082027553287953868142, 18.87024187847379721094465112707, 19.686435362865385611066149985375, 20.06071323234142809207235733850

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