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Properties

Label 1-1575-1575.1262-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.131 - 0.991i$
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  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)11-s + (0.743 − 0.669i)13-s + (0.669 + 0.743i)16-s + (−0.406 − 0.913i)17-s + (0.104 − 0.994i)19-s + (0.994 − 0.104i)22-s + (−0.951 + 0.309i)23-s + (0.5 + 0.866i)26-s + (0.913 + 0.406i)29-s + (−0.104 + 0.994i)31-s + (−0.866 + 0.5i)32-s + (0.978 − 0.207i)34-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)11-s + (0.743 − 0.669i)13-s + (0.669 + 0.743i)16-s + (−0.406 − 0.913i)17-s + (0.104 − 0.994i)19-s + (0.994 − 0.104i)22-s + (−0.951 + 0.309i)23-s + (0.5 + 0.866i)26-s + (0.913 + 0.406i)29-s + (−0.104 + 0.994i)31-s + (−0.866 + 0.5i)32-s + (0.978 − 0.207i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.131 - 0.991i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.131 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3431065316 - 0.3917126601i\)
\(L(\frac12)\) \(\approx\) \(0.3431065316 - 0.3917126601i\)
\(L(1)\) \(\approx\) \(0.7291742726 + 0.1351440834i\)
\(L(1)\) \(\approx\) \(0.7291742726 + 0.1351440834i\)

Euler product

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

−20.6134754656462627535933641700, −20.07865389040599929107958300617, −19.27582564216226096258838389049, −18.43749903021307508888321528891, −18.05816488921624040468299412304, −17.079413457396475850628304616486, −16.4840748082355540525859790294, −15.38149395493847366256093925251, −14.61393112947611892317072782200, −13.656912172617413255556316201939, −13.15391597062796959004737119649, −12.12892166078283529348026349278, −11.79278075758971599435488920541, −10.6838562109773017655122160606, −10.13006806049239368470190370186, −9.439155997216610634240836283912, −8.37103740531809061805797110633, −7.979501757820556531361112946473, −6.712393157368313654820299449229, −5.82018744723958285652124514942, −4.609642644841938315522584059936, −4.09742878155432322622259161628, −3.14190371364160115819333914821, −1.99305763907148300069592762035, −1.49534011737417318207232473461, 0.21621927808316458361049159899, 1.31612470552321354907804625591, 2.85163297969882371813885591073, 3.71424096797471174629730117996, 4.852760346197459148569535389558, 5.47122244885823339325511162536, 6.34493690327575606594587272612, 7.06084124020653368364932531918, 8.005444790882131528010986093984, 8.63519711186216163028101348838, 9.30933050012351656724718180833, 10.35306619024320049029690843887, 10.963929542061437216888328473716, 12.01700330909864097942690079368, 13.11471974893992877320356124301, 13.6766655627576372599370608062, 14.22765968726679025741854407765, 15.335091641628951793625492447760, 15.902752937420131148963373454699, 16.32169734640875900426769851263, 17.37655781814952363724055246923, 18.12766497473740548845379035629, 18.39974234903814890876862420979, 19.600552824304423795893037289, 20.00470745688388680977628583264

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