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Properties

Label 2-975-5.4-c1-0-4
Degree $2$
Conductor $975$
Sign $0.447 + 0.894i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  − 2.77i·2-s − i·3-s − 5.71·4-s − 2.77·6-s + 2.71i·7-s + 10.3i·8-s − 9-s − 2.71·11-s + 5.71i·12-s + i·13-s + 7.55·14-s + 17.2·16-s + 2.83i·17-s + 2.77i·18-s + 3.55·19-s + ⋯
L(s)  = 1  − 1.96i·2-s − 0.577i·3-s − 2.85·4-s − 1.13·6-s + 1.02i·7-s + 3.65i·8-s − 0.333·9-s − 0.820·11-s + 1.65i·12-s + 0.277i·13-s + 2.01·14-s + 4.31·16-s + 0.688i·17-s + 0.654i·18-s + 0.816·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743524 - 0.459523i\)
\(L(\frac12)\) \(\approx\) \(0.743524 - 0.459523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 + 2.77iT - 2T^{2} \)
7 \( 1 - 2.71iT - 7T^{2} \)
11 \( 1 + 2.71T + 11T^{2} \)
17 \( 1 - 2.83iT - 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 + 4.83iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 - 4.27iT - 37T^{2} \)
41 \( 1 - 2.83T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 - 11.5iT - 47T^{2} \)
53 \( 1 - 1.16iT - 53T^{2} \)
59 \( 1 - 2.11T + 59T^{2} \)
61 \( 1 - 6.60T + 61T^{2} \)
67 \( 1 + 1.88iT - 67T^{2} \)
71 \( 1 + 6.71T + 71T^{2} \)
73 \( 1 - 9.11iT - 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 2.11iT - 83T^{2} \)
89 \( 1 + 1.16T + 89T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−9.982273504058395895922097777198, −9.271397625433621818096538308096, −8.448843189378902312958144377285, −7.84256083903042122915534401826, −6.11704385069823679757555643530, −5.23827549455576444617684915849, −4.30039198709771846893514865854, −2.96467997143573872064997596684, −2.42298471586196053943453682593, −1.23611321405951556586516670284, 0.46982835383278973730882110018, 3.42225285713798905431475253852, 4.26732705592754898469084346296, 5.24675698141683186756169204556, 5.70269919139442202744505731648, 7.05177773414504190564659540936, 7.41683285203310310313027527155, 8.241082687892807732523634138988, 9.140699322993607204497120372423, 9.909704695376116867404801418199

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