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Properties

Label 2-605-55.4-c1-0-6
Degree $2$
Conductor $605$
Sign $-0.706 - 0.707i$
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  + (−0.471 + 0.649i)2-s + (−1.67 − 0.544i)3-s + (0.418 + 1.28i)4-s + (−0.145 − 2.23i)5-s + (1.14 − 0.830i)6-s + (0.563 − 0.182i)7-s + (−2.56 − 0.832i)8-s + (0.0810 + 0.0589i)9-s + (1.51 + 0.958i)10-s − 2.38i·12-s + (−1.05 + 1.45i)13-s + (−0.146 + 0.452i)14-s + (−0.969 + 3.81i)15-s + (−0.444 + 0.322i)16-s + (4.15 + 5.72i)17-s + (−0.0765 + 0.0248i)18-s + ⋯
L(s)  = 1  + (−0.333 + 0.459i)2-s + (−0.966 − 0.314i)3-s + (0.209 + 0.644i)4-s + (−0.0652 − 0.997i)5-s + (0.466 − 0.339i)6-s + (0.212 − 0.0691i)7-s + (−0.905 − 0.294i)8-s + (0.0270 + 0.0196i)9-s + (0.479 + 0.302i)10-s − 0.689i·12-s + (−0.292 + 0.402i)13-s + (−0.0392 + 0.120i)14-s + (−0.250 + 0.985i)15-s + (−0.111 + 0.0807i)16-s + (1.00 + 1.38i)17-s + (−0.0180 + 0.00585i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.706 - 0.707i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174148 + 0.420262i\)
\(L(\frac12)\) \(\approx\) \(0.174148 + 0.420262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.145 + 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (0.471 - 0.649i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.67 + 0.544i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.563 + 0.182i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.05 - 1.45i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.15 - 5.72i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.706 - 2.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.49iT - 23T^{2} \)
29 \( 1 + (-1.10 - 3.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.98 + 3.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.97 - 2.26i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.59 - 7.99i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.51iT - 43T^{2} \)
47 \( 1 + (-1.83 - 0.596i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.40 - 1.92i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0118 - 0.0364i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.78 - 2.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.79iT - 67T^{2} \)
71 \( 1 + (-9.54 + 6.93i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.48 - 2.10i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.66 + 2.66i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.49 + 4.80i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (3.15 - 4.34i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.21322793475119501841530656690, −10.05835614429966818754606880676, −8.992364606769518366996788981326, −8.253410539186698589747228108965, −7.52610033228522164752261823750, −6.39570077060005943122435385976, −5.76738744397764028016002498369, −4.64078175206314117319074425759, −3.43102256718566398019579065111, −1.47363374087099109251546769991, 0.32021179147524802481028477866, 2.21950643209470465817927411680, 3.35064433067580460555068232996, 5.12062167315555751876562570651, 5.56625617827032124864792951771, 6.66721446277371402256031822114, 7.46606129411140478608210184300, 8.840551952401394032491789801308, 9.918276130485452906114436555386, 10.39162040584175055324143902223

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