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Properties

Label 2-1575-105.83-c0-0-4
Degree $2$
Conductor $1575$
Sign $-0.374 - 0.927i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
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  + (1 + i)2-s + i·4-s + (−0.707 + 0.707i)7-s + 1.41i·11-s − 1.41·14-s + 16-s + (−1.41 + 1.41i)22-s + (−1 + i)23-s + (−0.707 − 0.707i)28-s + 1.41·29-s + (1 + i)32-s + (1.41 − 1.41i)37-s + (−1.41 − 1.41i)43-s − 1.41·44-s − 2·46-s + ⋯
L(s)  = 1  + (1 + i)2-s + i·4-s + (−0.707 + 0.707i)7-s + 1.41i·11-s − 1.41·14-s + 16-s + (−1.41 + 1.41i)22-s + (−1 + i)23-s + (−0.707 − 0.707i)28-s + 1.41·29-s + (1 + i)32-s + (1.41 − 1.41i)37-s + (−1.41 − 1.41i)43-s − 1.41·44-s − 2·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.751838313\)
\(L(\frac12)\) \(\approx\) \(1.751838313\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.815058115810703438957590670150, −9.045671556612517321104457729427, −7.914334908967090917684493587885, −7.29050977855640185738331891894, −6.46141937531656666784095877689, −5.87062846748685346960231179231, −4.99473716344798591472882486957, −4.24405567623656101125730488152, −3.26212981803990160417194471414, −1.99853543848525930454647359952, 1.09574911540786564503374128921, 2.66671426068583057859247343096, 3.28527492964716103587803528489, 4.15706526693404224822204476378, 4.92000633166331703893434148320, 6.09312313545987489851557351538, 6.56215842585024997409073439737, 7.976276606161320309392002487064, 8.466219262566417476036316629381, 9.889281528924600794794481440043

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