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Properties

Label 2-605-55.54-c2-0-85
Degree $2$
Conductor $605$
Sign $0.975 - 0.218i$
Analytic cond. $16.4850$
Root an. cond. $4.06017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.55·2-s + 1.52i·3-s + 8.61·4-s + (4.52 − 2.13i)5-s + 5.41i·6-s − 1.09·7-s + 16.3·8-s + 6.67·9-s + (16.0 − 7.57i)10-s + 13.1i·12-s + 0.874·13-s − 3.87·14-s + (3.25 + 6.89i)15-s + 23.7·16-s − 23.3·17-s + 23.7·18-s + ⋯
L(s)  = 1  + 1.77·2-s + 0.508i·3-s + 2.15·4-s + (0.904 − 0.426i)5-s + 0.902i·6-s − 0.155·7-s + 2.04·8-s + 0.741·9-s + (1.60 − 0.757i)10-s + 1.09i·12-s + 0.0672·13-s − 0.276·14-s + (0.216 + 0.459i)15-s + 1.48·16-s − 1.37·17-s + 1.31·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(16.4850\)
Root analytic conductor: \(4.06017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1),\ 0.975 - 0.218i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.076651950\)
\(L(\frac12)\) \(\approx\) \(6.076651950\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.52 + 2.13i)T \)
11 \( 1 \)
good2 \( 1 - 3.55T + 4T^{2} \)
3 \( 1 - 1.52iT - 9T^{2} \)
7 \( 1 + 1.09T + 49T^{2} \)
13 \( 1 - 0.874T + 169T^{2} \)
17 \( 1 + 23.3T + 289T^{2} \)
19 \( 1 + 25.3iT - 361T^{2} \)
23 \( 1 - 33.3iT - 529T^{2} \)
29 \( 1 - 16.8iT - 841T^{2} \)
31 \( 1 + 6.40T + 961T^{2} \)
37 \( 1 + 65.5iT - 1.36e3T^{2} \)
41 \( 1 - 38.7iT - 1.68e3T^{2} \)
43 \( 1 - 15.0T + 1.84e3T^{2} \)
47 \( 1 - 59.2iT - 2.20e3T^{2} \)
53 \( 1 - 23.6iT - 2.80e3T^{2} \)
59 \( 1 - 38.0T + 3.48e3T^{2} \)
61 \( 1 + 34.0iT - 3.72e3T^{2} \)
67 \( 1 + 43.7iT - 4.48e3T^{2} \)
71 \( 1 - 36.2T + 5.04e3T^{2} \)
73 \( 1 + 84.0T + 5.32e3T^{2} \)
79 \( 1 - 30.3iT - 6.24e3T^{2} \)
83 \( 1 + 136.T + 6.88e3T^{2} \)
89 \( 1 + 8.46T + 7.92e3T^{2} \)
97 \( 1 - 66.6iT - 9.40e3T^{2} \)
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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

−10.87362380733956194740072603282, −9.656786403494086830492270227416, −9.006676486250021561651741722024, −7.31258951880785722003016291818, −6.56251124567752924315730172155, −5.61020299252598277281252382332, −4.80626651449065548935229209370, −4.14151162028222470796962875214, −2.90692834462835469562796800738, −1.74420054014722229446402180051, 1.76580210812563179098109737580, 2.59900584066778181036763888329, 3.88443938409850613850862057892, 4.79648857078810827236577205384, 5.92255036889312742364670806048, 6.55328397328201902958660771938, 7.12634629216650389894065011023, 8.484453864913953820542359233230, 9.945410547072442893041337662568, 10.57853060482283733069117677923

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