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Properties

Label 2-213-71.25-c1-0-9
Degree $2$
Conductor $213$
Sign $0.230 + 0.973i$
Analytic cond. $1.70081$
Root an. cond. $1.30415$
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  + (1.30 − 0.951i)2-s + (0.809 − 0.587i)3-s + (0.190 − 0.587i)4-s + (−0.881 − 2.71i)5-s + (0.5 − 1.53i)6-s + (1.30 − 0.951i)7-s + (0.690 + 2.12i)8-s + (0.309 − 0.951i)9-s + (−3.73 − 2.71i)10-s + (−4.92 + 3.57i)11-s + (−0.190 − 0.587i)12-s + (1.80 + 1.31i)13-s + (0.809 − 2.48i)14-s + (−2.30 − 1.67i)15-s + (3.92 + 2.85i)16-s + ⋯
L(s)  = 1  + (0.925 − 0.672i)2-s + (0.467 − 0.339i)3-s + (0.0954 − 0.293i)4-s + (−0.394 − 1.21i)5-s + (0.204 − 0.628i)6-s + (0.494 − 0.359i)7-s + (0.244 + 0.751i)8-s + (0.103 − 0.317i)9-s + (−1.18 − 0.858i)10-s + (−1.48 + 1.07i)11-s + (−0.0551 − 0.169i)12-s + (0.501 + 0.364i)13-s + (0.216 − 0.665i)14-s + (−0.596 − 0.433i)15-s + (0.981 + 0.713i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(213\)    =    \(3 \cdot 71\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(1.70081\)
Root analytic conductor: \(1.30415\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{213} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 213,\ (\ :1/2),\ 0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58548 - 1.25357i\)
\(L(\frac12)\) \(\approx\) \(1.58548 - 1.25357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (1.95 + 8.19i)T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.881 + 2.71i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.30 + 0.951i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.92 - 3.57i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.80 - 1.31i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-5.73 + 4.16i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (-2.88 - 8.86i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.04 - 4.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 - 3.09T + 41T^{2} \)
43 \( 1 + (0.118 + 0.363i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (5.42 + 3.94i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.89 + 12.0i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.80 - 5.56i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.16 + 3.75i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.472 - 1.45i)T + (-54.2 - 39.3i)T^{2} \)
73 \( 1 + (-8.35 + 6.06i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.600 + 1.84i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.69 + 11.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.972 + 2.99i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + 5.94T + 97T^{2} \)
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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

−12.45374681817448949435612613513, −11.52218705123654231889505442844, −10.46347974850906625940332532109, −9.052268929186346132556214708203, −8.112208082405362211383533337807, −7.31032489314366304360339425940, −5.14734145331997360538862683545, −4.69687698929824595521888919286, −3.29982979165322263147623790740, −1.74113079865308311380341710844, 2.88436744098542462967461381731, 3.85489758778458500192809170725, 5.37040867994625547979770983517, 6.08664674409370473602490251731, 7.59972062186585577381821141652, 8.081405341052435696004478041412, 9.806553925271356087090150026629, 10.65654900807613681066609058666, 11.52774413193642755060859208621, 12.92013808765077464232746181186

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