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Properties

Label 2-1575-21.5-c1-0-32
Degree $2$
Conductor $1575$
Sign $0.160 - 0.987i$
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  + (2.22 + 1.28i)2-s + (2.30 + 3.98i)4-s + (−0.151 − 2.64i)7-s + 6.68i·8-s + (4.88 − 2.81i)11-s + 3.53i·13-s + (3.05 − 6.07i)14-s + (−3.98 + 6.90i)16-s + (2.67 + 4.62i)17-s + (1.24 + 0.721i)19-s + 14.4·22-s + (−2.59 − 1.49i)23-s + (−4.54 + 7.86i)26-s + (10.1 − 6.68i)28-s + 2.19i·29-s + ⋯
L(s)  = 1  + (1.57 + 0.908i)2-s + (1.15 + 1.99i)4-s + (−0.0573 − 0.998i)7-s + 2.36i·8-s + (1.47 − 0.849i)11-s + 0.980i·13-s + (0.816 − 1.62i)14-s + (−0.996 + 1.72i)16-s + (0.647 + 1.12i)17-s + (0.286 + 0.165i)19-s + 3.08·22-s + (−0.541 − 0.312i)23-s + (−0.890 + 1.54i)26-s + (1.92 − 1.26i)28-s + 0.407i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.160 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.599822106\)
\(L(\frac12)\) \(\approx\) \(4.599822106\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.151 + 2.64i)T \)
good2 \( 1 + (-2.22 - 1.28i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-4.88 + 2.81i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.53iT - 13T^{2} \)
17 \( 1 + (-2.67 - 4.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.24 - 0.721i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.19iT - 29T^{2} \)
31 \( 1 + (-1.31 + 0.761i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.546 - 0.946i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.52T + 41T^{2} \)
43 \( 1 - 0.486T + 43T^{2} \)
47 \( 1 + (2.12 - 3.68i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.48 + 2.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.70 + 9.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.06 - 3.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.24 + 2.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (2.71 - 1.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.40 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.77T + 83T^{2} \)
89 \( 1 + (5.16 - 8.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.7iT - 97T^{2} \)
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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

−9.503782375295197675205225247144, −8.446464583794993369080536261608, −7.76413419564897251320358070927, −6.68650112395887588873839602889, −6.50316462792876377334780694603, −5.57686868006603438858237701118, −4.47080574067456957597242091205, −3.86776968688853235639282348252, −3.29702018155001139320903113940, −1.55143081874051364384284682523, 1.29940624419380209903792351781, 2.39918687159361890805456172676, 3.22087037726722647842282521429, 4.10018063818980716899429884453, 5.05890758852029898452790272383, 5.63143428225782467408084209568, 6.47175376355331732135847610520, 7.33962637614355359250552580822, 8.627917008016240737591742433821, 9.681529349468383756965490021863

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