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Properties

Label 2-605-55.14-c1-0-19
Degree $2$
Conductor $605$
Sign $0.936 - 0.351i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (0.192 + 0.265i)2-s + (0.914 − 0.297i)3-s + (0.584 − 1.79i)4-s + (−0.789 + 2.09i)5-s + (0.255 + 0.185i)6-s + (3.11 + 1.01i)7-s + (1.21 − 0.394i)8-s + (−1.67 + 1.21i)9-s + (−0.708 + 0.194i)10-s − 1.82i·12-s + (3.06 + 4.21i)13-s + (0.332 + 1.02i)14-s + (−0.100 + 2.14i)15-s + (−2.72 − 1.97i)16-s + (2.11 − 2.91i)17-s + (−0.647 − 0.210i)18-s + ⋯
L(s)  = 1  + (0.136 + 0.187i)2-s + (0.528 − 0.171i)3-s + (0.292 − 0.899i)4-s + (−0.352 + 0.935i)5-s + (0.104 + 0.0757i)6-s + (1.17 + 0.382i)7-s + (0.429 − 0.139i)8-s + (−0.559 + 0.406i)9-s + (−0.223 + 0.0613i)10-s − 0.525i·12-s + (0.848 + 1.16i)13-s + (0.0887 + 0.273i)14-s + (−0.0258 + 0.554i)15-s + (−0.680 − 0.494i)16-s + (0.513 − 0.706i)17-s + (−0.152 − 0.0496i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.936 - 0.351i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.936 - 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08679 + 0.378888i\)
\(L(\frac12)\) \(\approx\) \(2.08679 + 0.378888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.789 - 2.09i)T \)
11 \( 1 \)
good2 \( 1 + (-0.192 - 0.265i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.914 + 0.297i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-3.11 - 1.01i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.06 - 4.21i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.11 + 2.91i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.54 - 4.76i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.84iT - 23T^{2} \)
29 \( 1 + (0.632 - 1.94i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.63 + 1.91i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.93 - 1.92i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.15 + 9.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.596iT - 43T^{2} \)
47 \( 1 + (-0.914 + 0.297i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.231 - 0.318i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.162 + 0.501i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.30 + 3.13i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.45iT - 67T^{2} \)
71 \( 1 + (10.1 + 7.36i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.09 - 2.63i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.29 - 1.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.13 - 9.82i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (6.63 + 9.13i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.87253308234327988116278192768, −9.937276586675702910168044765876, −8.800561799822032094507741394112, −8.015757255371466963603967603877, −7.16034258606921564117477753184, −6.19023890513073372901488603021, −5.29951470373885516079909394413, −4.13371774866365631854078849172, −2.66392144312839423717510934055, −1.67349441576243029302247601957, 1.29884311802935390901337389556, 2.97633701459526206302923666321, 3.84377307324893343084111253814, 4.79884065979223998340475922426, 5.90875558057938044816237593620, 7.50890385690222966692431274062, 8.114952817704524504235320876685, 8.532101070969312931500931393456, 9.554657400801137076084123568392, 10.92195395837975726113798983483

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