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Properties

Label 2-975-195.77-c1-0-2
Degree $2$
Conductor $975$
Sign $-0.924 + 0.379i$
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  + (0.299 + 0.299i)2-s + (0.125 + 1.72i)3-s − 1.82i·4-s + (−0.479 + 0.554i)6-s + (−0.976 + 0.976i)7-s + (1.14 − 1.14i)8-s + (−2.96 + 0.431i)9-s − 1.78·11-s + (3.14 − 0.227i)12-s + (−2.75 − 2.32i)13-s − 0.584·14-s − 2.95·16-s + (−3.26 + 3.26i)17-s + (−1.01 − 0.759i)18-s − 6.18·19-s + ⋯
L(s)  = 1  + (0.211 + 0.211i)2-s + (0.0721 + 0.997i)3-s − 0.910i·4-s + (−0.195 + 0.226i)6-s + (−0.368 + 0.368i)7-s + (0.404 − 0.404i)8-s + (−0.989 + 0.143i)9-s − 0.537·11-s + (0.908 − 0.0657i)12-s + (−0.765 − 0.643i)13-s − 0.156·14-s − 0.739·16-s + (−0.790 + 0.790i)17-s + (−0.239 − 0.178i)18-s − 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.924 + 0.379i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.924 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0301231 - 0.152601i\)
\(L(\frac12)\) \(\approx\) \(0.0301231 - 0.152601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.125 - 1.72i)T \)
5 \( 1 \)
13 \( 1 + (2.75 + 2.32i)T \)
good2 \( 1 + (-0.299 - 0.299i)T + 2iT^{2} \)
7 \( 1 + (0.976 - 0.976i)T - 7iT^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
17 \( 1 + (3.26 - 3.26i)T - 17iT^{2} \)
19 \( 1 + 6.18T + 19T^{2} \)
23 \( 1 + (-0.696 - 0.696i)T + 23iT^{2} \)
29 \( 1 - 7.33T + 29T^{2} \)
31 \( 1 - 6.61iT - 31T^{2} \)
37 \( 1 + (5.21 - 5.21i)T - 37iT^{2} \)
41 \( 1 + 6.45T + 41T^{2} \)
43 \( 1 + (-1.78 + 1.78i)T - 43iT^{2} \)
47 \( 1 + (5.69 + 5.69i)T + 47iT^{2} \)
53 \( 1 + (1.74 + 1.74i)T + 53iT^{2} \)
59 \( 1 + 9.15iT - 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 + (3.72 - 3.72i)T - 67iT^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + (-1.35 - 1.35i)T + 73iT^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + (-7.93 + 7.93i)T - 83iT^{2} \)
89 \( 1 + 3.75iT - 89T^{2} \)
97 \( 1 + (-6.92 + 6.92i)T - 97iT^{2} \)
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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

−10.40799302059570334526070132607, −9.878865175854569004376088822480, −8.838117170834551897596530996370, −8.269094243413640020637446418958, −6.77841705318171533120909168716, −6.11328285141204490137506352101, −5.08110115131388128524382191600, −4.63513575908265847467316704220, −3.31364463988422677234740759802, −2.14765681512664996195815228836, 0.05940032456887405762153628382, 2.14242524685791498467689178742, 2.80366030495277644985307621937, 4.08815363508976846330322353165, 5.01295494918612545981721917550, 6.48627682797197680524953120470, 6.96388705097467554354472860401, 7.81718844720653978241947460123, 8.536588287572663786081891952781, 9.354867719373889534775941414254

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