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Properties

Label 2-1575-35.34-c0-0-6
Degree $2$
Conductor $1575$
Sign $-0.447 + 0.894i$
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  − i·2-s − i·7-s − i·8-s + 11-s − 14-s − 16-s − i·22-s + i·23-s − 29-s + i·37-s − i·43-s + 46-s − 49-s − 2i·53-s − 56-s + ⋯
L(s)  = 1  − i·2-s − i·7-s − i·8-s + 11-s − 14-s − 16-s − i·22-s + i·23-s − 29-s + i·37-s − i·43-s + 46-s − 49-s − 2i·53-s − 56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273977868\)
\(L(\frac12)\) \(\approx\) \(1.273977868\)
\(L(1)\) 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 \( 1 + iT \)
good2 \( 1 + iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.697831863137748353464176588330, −8.808036408677867291682335091912, −7.66476373088840608831281575575, −6.96840610634510930146297951669, −6.27966694123544293748218645752, −5.00286332828853934429028089546, −3.82671895408862407941536656031, −3.51480041582539475256755452809, −2.08086484378021364477569432625, −1.09915149154151201136701894336, 1.84764916556910467273797050135, 2.87465698894886530916011711200, 4.20615326396593357826942978377, 5.21546728565667259673098134671, 6.01379839869926721671148843739, 6.52929929537319611138161596603, 7.42557907845317365135588293843, 8.205357190835298406048859052435, 8.976846395961361110782148213354, 9.477065285462139530277909108834

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