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Properties

Label 2-1275-85.84-c1-0-2
Degree $2$
Conductor $1275$
Sign $0.133 + 0.991i$
Analytic cond. $10.1809$
Root an. cond. $3.19075$
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  + 2.67i·2-s − 3-s − 5.14·4-s − 2.67i·6-s + 3.95·7-s − 8.39i·8-s + 9-s + 2.28i·11-s + 5.14·12-s + 1.90i·13-s + 10.5i·14-s + 12.1·16-s + (−2.31 − 3.40i)17-s + 2.67i·18-s − 6.58·19-s + ⋯
L(s)  = 1  + 1.88i·2-s − 0.577·3-s − 2.57·4-s − 1.09i·6-s + 1.49·7-s − 2.96i·8-s + 0.333·9-s + 0.689i·11-s + 1.48·12-s + 0.527i·13-s + 2.82i·14-s + 3.03·16-s + (−0.562 − 0.826i)17-s + 0.629i·18-s − 1.51·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.133 + 0.991i$
Analytic conductor: \(10.1809\)
Root analytic conductor: \(3.19075\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1275,\ (\ :1/2),\ 0.133 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4683222236\)
\(L(\frac12)\) \(\approx\) \(0.4683222236\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
17 \( 1 + (2.31 + 3.40i)T \)
good2 \( 1 - 2.67iT - 2T^{2} \)
7 \( 1 - 3.95T + 7T^{2} \)
11 \( 1 - 2.28iT - 11T^{2} \)
13 \( 1 - 1.90iT - 13T^{2} \)
19 \( 1 + 6.58T + 19T^{2} \)
23 \( 1 + 7.11T + 23T^{2} \)
29 \( 1 - 2.23iT - 29T^{2} \)
31 \( 1 - 9.45iT - 31T^{2} \)
37 \( 1 + 6.57T + 37T^{2} \)
41 \( 1 + 1.52iT - 41T^{2} \)
43 \( 1 - 4.26iT - 43T^{2} \)
47 \( 1 + 3.61iT - 47T^{2} \)
53 \( 1 + 1.48iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 15.0iT - 61T^{2} \)
67 \( 1 + 4.91iT - 67T^{2} \)
71 \( 1 + 9.15iT - 71T^{2} \)
73 \( 1 - 7.94T + 73T^{2} \)
79 \( 1 - 3.92iT - 79T^{2} \)
83 \( 1 + 3.85iT - 83T^{2} \)
89 \( 1 - 4.71T + 89T^{2} \)
97 \( 1 + 7.70T + 97T^{2} \)
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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

−10.13140350559918712187170101495, −9.035395344559115611194275221172, −8.484121505082716792325928579351, −7.68742928552838957021151212193, −6.95708568469706470117698405735, −6.33108897614731870431311652647, −5.28046553356398521325424420684, −4.71155515140613551495444243984, −4.13212963378144808924825769043, −1.75786402135501101210817231407, 0.21512678254384284887278644242, 1.63126822085772833700185833170, 2.33156504911571404396102036005, 3.87323194306702683297771751800, 4.36488385972982692674683305885, 5.32825470273483015450305217220, 6.19475045820600026177680256555, 8.084586203463641118387980766842, 8.243921573256570063496851118784, 9.308315481455694046764697799627

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