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Properties

Label 2-605-55.52-c1-0-8
Degree $2$
Conductor $605$
Sign $0.813 + 0.580i$
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  + (−2.71 − 1.38i)3-s + (1.90 + 0.618i)4-s + (−2.22 + 0.238i)5-s + (3.71 + 5.11i)9-s + (−4.31 − 4.31i)12-s + (6.37 + 2.43i)15-s + (3.23 + 2.35i)16-s + (−4.37 − 0.919i)20-s + (2.84 − 2.84i)23-s + (4.88 − 1.06i)25-s + (−1.58 − 9.99i)27-s + (8.04 − 5.84i)31-s + (3.90 + 12.0i)36-s + (1.85 − 0.946i)37-s + (−9.47 − 10.4i)45-s + ⋯
L(s)  = 1  + (−1.57 − 0.800i)3-s + (0.951 + 0.309i)4-s + (−0.994 + 0.106i)5-s + (1.23 + 1.70i)9-s + (−1.24 − 1.24i)12-s + (1.64 + 0.627i)15-s + (0.809 + 0.587i)16-s + (−0.978 − 0.205i)20-s + (0.592 − 0.592i)23-s + (0.977 − 0.212i)25-s + (−0.304 − 1.92i)27-s + (1.44 − 1.05i)31-s + (0.650 + 2.00i)36-s + (0.305 − 0.155i)37-s + (−1.41 − 1.56i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.813 + 0.580i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.813 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851389 - 0.272648i\)
\(L(\frac12)\) \(\approx\) \(0.851389 - 0.272648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 - 0.238i)T \)
11 \( 1 \)
good2 \( 1 + (-1.90 - 0.618i)T^{2} \)
3 \( 1 + (2.71 + 1.38i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (-4.11 + 5.66i)T^{2} \)
13 \( 1 + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.84 + 2.84i)T - 23iT^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.04 + 5.84i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.85 + 0.946i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-5.98 + 11.7i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (0.803 - 5.07i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (3.15 + 1.02i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-11.4 - 11.4i)T + 67iT^{2} \)
71 \( 1 + (-2.42 - 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-78.9 + 25.6i)T^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (-4.92 - 0.779i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.91525953022035179406140414510, −10.19870044920340488560825029396, −8.433796670280209337897505093682, −7.58043065837780034167870574811, −6.91567402238349283163646110396, −6.27710960400480598805866540270, −5.23196255371594144200458767662, −4.03416058947678392276392503342, −2.46756562880154215518287358199, −0.832662825844964803696005064856, 0.981472322976162217331389015377, 3.16953253624133243027467096114, 4.40226497232965114708117786622, 5.20788178507844311477266988504, 6.18903929282754787637890807671, 6.91517179816745322659284611848, 7.87931522433873842878254658707, 9.270842825841395828212725908259, 10.26059966217627774420561411100, 10.89990985823510526114184061975

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