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Properties

Label 2-1575-21.17-c1-0-0
Degree $2$
Conductor $1575$
Sign $-0.736 + 0.675i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)4-s + (−2.29 + 1.32i)7-s + (2.44 + 1.41i)11-s − 2.64i·13-s + (−1.99 − 3.46i)16-s + (−1.87 + 3.24i)17-s + (−1.5 + 0.866i)19-s + (−5.61 + 3.24i)23-s − 5.29i·28-s − 1.41i·29-s + (4.5 + 2.59i)31-s + (−2.29 − 3.96i)37-s − 4.89·41-s + 4.58·43-s + (−4.89 + 2.82i)44-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.866 + 0.499i)7-s + (0.738 + 0.426i)11-s − 0.733i·13-s + (−0.499 − 0.866i)16-s + (−0.453 + 0.785i)17-s + (−0.344 + 0.198i)19-s + (−1.17 + 0.675i)23-s − 0.999i·28-s − 0.262i·29-s + (0.808 + 0.466i)31-s + (−0.376 − 0.652i)37-s − 0.765·41-s + 0.698·43-s + (−0.738 + 0.426i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.736 + 0.675i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.736 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08353974293\)
\(L(\frac12)\) \(\approx\) \(0.08353974293\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.29 - 1.32i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.64iT - 13T^{2} \)
17 \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.61 - 3.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.29 + 3.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 4.58T + 43T^{2} \)
47 \( 1 + (1.87 + 3.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.2 + 6.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 + 6.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9 - 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.87 + 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (11.4 + 6.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + (-8.57 - 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.29iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.758541096947369551543549623711, −9.099840347740036269611052295846, −8.356234310916696128093061430261, −7.67584516874304923027213891621, −6.64199906276305895457126173231, −5.98580571635552947867244579149, −4.84288383186734565886158242802, −3.85998400104790724532524777688, −3.23396844679688110543941656668, −1.99470829284800251806931392972, 0.03371234082880158864064672573, 1.34185492702781532146101915418, 2.75168860840055130154344214741, 4.07450329971033821321769592486, 4.54510696923626773854426044032, 5.82486277594861202323142950198, 6.45213230275450733053185184230, 7.06407542364237954357301587219, 8.378169537836345987586798091955, 9.066704065980680437630567442366

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