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Properties

Label 2-605-11.3-c1-0-30
Degree $2$
Conductor $605$
Sign $-0.970 - 0.242i$
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  + (−0.809 + 0.587i)2-s + (−0.927 − 2.85i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (2.42 + 1.76i)6-s + (0.927 − 2.85i)7-s + (−0.927 − 2.85i)8-s + (−4.85 + 3.52i)9-s + 10-s + 2.99·12-s + (3.23 − 2.35i)13-s + (0.927 + 2.85i)14-s + (−0.927 + 2.85i)15-s + (0.809 + 0.587i)16-s + (1.85 − 5.70i)18-s + (−1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.535 − 1.64i)3-s + (−0.154 + 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.990 + 0.719i)6-s + (0.350 − 1.07i)7-s + (−0.327 − 1.00i)8-s + (−1.61 + 1.17i)9-s + 0.316·10-s + 0.866·12-s + (0.897 − 0.652i)13-s + (0.247 + 0.762i)14-s + (−0.239 + 0.736i)15-s + (0.202 + 0.146i)16-s + (0.437 − 1.34i)18-s + (−0.283 − 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0429491 + 0.349389i\)
\(L(\frac12)\) \(\approx\) \(0.0429491 + 0.349389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.927 + 2.85i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.927 + 2.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.23 + 2.35i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.47 - 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.54 - 4.75i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (0.927 + 2.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.23 - 2.35i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.618 - 1.90i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.89 + 6.46i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (1.61 + 1.17i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.47 + 7.60i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.09 + 5.87i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.23 - 2.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (-6.47 + 4.70i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.28560307454013654694560326063, −8.863199845340456103435549796597, −8.012704279504622753404445195882, −7.68526877783458420497032390540, −6.79576398921989384411615240376, −6.10462450170144042641760949240, −4.62321145103935379412868272186, −3.31137189273840002909054879842, −1.43357776583375152948169618928, −0.26532395216767689579197965999, 2.09322317775140740705668739360, 3.68517248052460896578672336191, 4.55671252444591504234826754666, 5.70677621803806252157829683942, 6.07876535086056119754467285820, 8.117333152300158544616671906262, 8.851699249047911806387221722435, 9.515941128877154961800980708495, 10.30088890942140638558860502878, 10.89430812141462128104851938165

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