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Properties

Label 2-605-55.9-c1-0-17
Degree $2$
Conductor $605$
Sign $0.833 + 0.552i$
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  + (−0.763 − 0.248i)2-s + (1.03 + 1.42i)3-s + (−1.09 − 0.796i)4-s + (−2.16 − 0.550i)5-s + (−0.436 − 1.34i)6-s + (−0.348 + 0.479i)7-s + (1.58 + 2.17i)8-s + (−0.0309 + 0.0953i)9-s + (1.51 + 0.958i)10-s − 2.38i·12-s + (−1.70 − 0.554i)13-s + (0.384 − 0.279i)14-s + (−1.45 − 3.65i)15-s + (0.169 + 0.522i)16-s + (6.73 − 2.18i)17-s + (0.0472 − 0.0650i)18-s + ⋯
L(s)  = 1  + (−0.539 − 0.175i)2-s + (0.597 + 0.822i)3-s + (−0.548 − 0.398i)4-s + (−0.969 − 0.246i)5-s + (−0.178 − 0.548i)6-s + (−0.131 + 0.181i)7-s + (0.559 + 0.770i)8-s + (−0.0103 + 0.0317i)9-s + (0.479 + 0.302i)10-s − 0.689i·12-s + (−0.473 − 0.153i)13-s + (0.102 − 0.0746i)14-s + (−0.376 − 0.944i)15-s + (0.0424 + 0.130i)16-s + (1.63 − 0.530i)17-s + (0.0111 − 0.0153i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.833 + 0.552i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.833 + 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898767 - 0.270666i\)
\(L(\frac12)\) \(\approx\) \(0.898767 - 0.270666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.16 + 0.550i)T \)
11 \( 1 \)
good2 \( 1 + (0.763 + 0.248i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.03 - 1.42i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.348 - 0.479i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.70 + 0.554i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-6.73 + 2.18i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.85 + 1.34i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.49iT - 23T^{2} \)
29 \( 1 + (2.89 + 2.10i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.90 + 5.86i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.31 + 5.93i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.80 + 4.94i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.51iT - 43T^{2} \)
47 \( 1 + (1.13 + 1.56i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.26 + 0.736i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.0309 + 0.0224i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.06 - 3.27i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.79iT - 67T^{2} \)
71 \( 1 + (3.64 + 11.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.01 + 5.51i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.39 + 4.30i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.65 - 1.83i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (5.11 + 1.66i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.30214085094211345010951042309, −9.482594211715218873607139306370, −9.170365526125631931047492212205, −8.039561536777671487016088684436, −7.52485410472445257857151331795, −5.80070767202831297495273283094, −4.75617625978447502506824896219, −3.97207082854104150180565552118, −2.84271213613437343061923868648, −0.75011902987161056402503347395, 1.17554990902077992120733454787, 3.01742367822913839278733653246, 3.87920156760399832556852254328, 5.12229412785778187565355702044, 6.76294642897586140298811918554, 7.55002020858496901124488896911, 7.933780213845379783667417739951, 8.675317404662125486654603506815, 9.761738027357276571296872320874, 10.51578379901778534747440274574

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