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Properties

Label 2-605-11.9-c1-0-18
Degree $2$
Conductor $605$
Sign $0.998 + 0.0475i$
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  + (0.309 − 0.951i)2-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (2.42 − 1.76i)8-s + (−0.927 + 2.85i)9-s + 0.999·10-s + (0.618 − 1.90i)13-s + (−0.309 − 0.951i)16-s + (1.85 + 5.70i)17-s + (2.42 + 1.76i)18-s + (3.23 − 2.35i)19-s + (−0.309 + 0.951i)20-s + 4·23-s + (−0.809 + 0.587i)25-s + (−1.61 − 1.17i)26-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.404 + 0.293i)4-s + (0.138 + 0.425i)5-s + (0.858 − 0.623i)8-s + (−0.309 + 0.951i)9-s + 0.316·10-s + (0.171 − 0.527i)13-s + (−0.0772 − 0.237i)16-s + (0.449 + 1.38i)17-s + (0.572 + 0.415i)18-s + (0.742 − 0.539i)19-s + (−0.0690 + 0.212i)20-s + 0.834·23-s + (−0.161 + 0.117i)25-s + (−0.317 − 0.230i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.998 + 0.0475i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00680 - 0.0477185i\)
\(L(\frac12)\) \(\approx\) \(2.00680 - 0.0477185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.618 + 1.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.85 - 5.70i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (4.85 + 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.47 - 7.60i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.61 - 1.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.61 - 1.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-9.70 + 7.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.618 - 1.90i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.23 + 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.09 + 9.51i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + (-2.47 - 7.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (11.3 + 8.22i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.47 + 7.60i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.23 + 3.80i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-3.09 + 9.51i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.66166102681131006910536279441, −10.26408761554736292702178467863, −8.931274868894115734583602160274, −7.85852316944260860190127427644, −7.24034142352014605276493786027, −6.06534476430725092683502289236, −5.01550938435220264605097937339, −3.67600752864921621911526212921, −2.82436215237321859409327931904, −1.65882273118117894236626536074, 1.21360986963608547360533745730, 2.83745510498424125976544965647, 4.26266761787728113274926415186, 5.43689252018902517298631034226, 5.99978505123612106977968025759, 7.11920921342202300206235625869, 7.68714070253197844306683202530, 9.097834744883504177127994029202, 9.487583686940494100183068150335, 10.75366531519943379233813965571

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