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Properties

Label 2-605-11.4-c1-0-35
Degree $2$
Conductor $605$
Sign $-0.138 - 0.990i$
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  + (−1.11 − 0.806i)2-s + (1.00 − 3.10i)3-s + (−0.0357 − 0.110i)4-s + (−0.809 + 0.587i)5-s + (−3.62 + 2.63i)6-s + (−0.424 − 1.30i)7-s + (−0.897 + 2.76i)8-s + (−6.18 − 4.49i)9-s + 1.37·10-s − 0.377·12-s + (1.56 + 1.13i)13-s + (−0.582 + 1.79i)14-s + (1.00 + 3.10i)15-s + (3.03 − 2.20i)16-s + (−5.02 + 3.64i)17-s + (3.24 + 9.97i)18-s + ⋯
L(s)  = 1  + (−0.785 − 0.570i)2-s + (0.581 − 1.79i)3-s + (−0.0178 − 0.0550i)4-s + (−0.361 + 0.262i)5-s + (−1.47 + 1.07i)6-s + (−0.160 − 0.493i)7-s + (−0.317 + 0.976i)8-s + (−2.06 − 1.49i)9-s + 0.434·10-s − 0.108·12-s + (0.433 + 0.315i)13-s + (−0.155 + 0.478i)14-s + (0.260 + 0.800i)15-s + (0.759 − 0.551i)16-s + (−1.21 + 0.885i)17-s + (0.763 + 2.35i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325882 + 0.374722i\)
\(L(\frac12)\) \(\approx\) \(0.325882 + 0.374722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (1.11 + 0.806i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.00 + 3.10i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.424 + 1.30i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.56 - 1.13i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.02 - 3.64i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.251 + 0.773i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 + (2.42 + 7.45i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.75 - 2.00i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.39 + 4.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.564 + 1.73i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.46T + 43T^{2} \)
47 \( 1 + (-1.46 + 4.50i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.79 - 4.93i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.344 - 1.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.05 + 1.49i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.73T + 67T^{2} \)
71 \( 1 + (-0.902 + 0.655i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.83 + 11.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.69 + 3.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.9 - 9.39i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 + (11.4 + 8.32i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.00242952233406842573683991439, −8.833885350928758923449280939392, −8.400456135637043385904480081880, −7.51060333112978270571299579490, −6.64928937414927218839924345918, −5.85965272933249061145028952630, −3.92191710103847471186708270557, −2.51506754504420344328524071503, −1.68083470957801667224890728827, −0.32392222574793702142182010774, 2.83082127440558557276706804051, 3.78954736056407559826933243435, 4.65772931241061378483034584836, 5.74670324024918677370993655885, 7.08516480066090505440902396284, 8.367051371126793501169852115497, 8.592948294883232539984241147045, 9.435436969354648807417839299510, 10.00213759440850233999170091398, 10.98481418784557873808144767585

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