Properties

Label 2-605-11.3-c1-0-15
Degree $2$
Conductor $605$
Sign $-0.220 - 0.975i$
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  + (−1.69 + 1.23i)2-s + (0.591 + 1.81i)3-s + (0.738 − 2.27i)4-s + (0.809 + 0.587i)5-s + (−3.24 − 2.35i)6-s + (0.947 − 2.91i)7-s + (0.252 + 0.777i)8-s + (−0.533 + 0.387i)9-s − 2.09·10-s + 4.57·12-s + (2.46 − 1.79i)13-s + (1.98 + 6.11i)14-s + (−0.591 + 1.81i)15-s + (2.48 + 1.80i)16-s + (−0.375 − 0.272i)17-s + (0.427 − 1.31i)18-s + ⋯
L(s)  = 1  + (−1.19 + 0.870i)2-s + (0.341 + 1.05i)3-s + (0.369 − 1.13i)4-s + (0.361 + 0.262i)5-s + (−1.32 − 0.961i)6-s + (0.358 − 1.10i)7-s + (0.0893 + 0.274i)8-s + (−0.177 + 0.129i)9-s − 0.662·10-s + 1.31·12-s + (0.684 − 0.497i)13-s + (0.530 + 1.63i)14-s + (−0.152 + 0.469i)15-s + (0.620 + 0.450i)16-s + (−0.0910 − 0.0661i)17-s + (0.100 − 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 605,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650666 + 0.814438i\)
\(L(\frac12)\) \(\approx\) \(0.650666 + 0.814438i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 1.23i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.591 - 1.81i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.947 + 2.91i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.46 + 1.79i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.375 + 0.272i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.43 - 7.50i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 + (1.15 - 3.53i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.47 + 6.15i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.569 + 1.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.36 - 4.19i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 + (0.920 + 2.83i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.38 - 2.46i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.869 + 2.67i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.63 + 1.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + (-5.27 - 3.83i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.05 + 9.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.35 - 6.79i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.21 - 5.24i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (12.4 - 9.02i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.30152216571974575130929345014, −10.02993616071349080116896203395, −9.231821678861424890275486154184, −8.202488777717575328716032222980, −7.67228856989515025613625223403, −6.59952465341945191767375312018, −5.68218452664452406521191840978, −4.26955004033676091953034830091, −3.40595830989253007098971557085, −1.21740400565484983719924901005, 1.08286063106032696167277620798, 2.09612114075353792966860797197, 2.84336782846742766736340212573, 4.83206980394958603056895414799, 6.07672991243410218430802639577, 7.12962037987093702667935362466, 8.165320740431770563789497740419, 8.733059955866788460833320996807, 9.325415246022063331503876318246, 10.32470109886809812951353022337

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