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Properties

Label 2-605-55.14-c1-0-36
Degree $2$
Conductor $605$
Sign $-0.178 + 0.983i$
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  + (−0.471 − 0.649i)2-s + (1.67 − 0.544i)3-s + (0.418 − 1.28i)4-s + (2.07 − 0.828i)5-s + (−1.14 − 0.830i)6-s + (0.563 + 0.182i)7-s + (−2.56 + 0.832i)8-s + (0.0810 − 0.0589i)9-s + (−1.51 − 0.958i)10-s − 2.38i·12-s + (−1.05 − 1.45i)13-s + (−0.146 − 0.452i)14-s + (3.02 − 2.51i)15-s + (−0.444 − 0.322i)16-s + (4.15 − 5.72i)17-s + (−0.0765 − 0.0248i)18-s + ⋯
L(s)  = 1  + (−0.333 − 0.459i)2-s + (0.966 − 0.314i)3-s + (0.209 − 0.644i)4-s + (0.928 − 0.370i)5-s + (−0.466 − 0.339i)6-s + (0.212 + 0.0691i)7-s + (−0.905 + 0.294i)8-s + (0.0270 − 0.0196i)9-s + (−0.479 − 0.302i)10-s − 0.689i·12-s + (−0.292 − 0.402i)13-s + (−0.0392 − 0.120i)14-s + (0.781 − 0.649i)15-s + (−0.111 − 0.0807i)16-s + (1.00 − 1.38i)17-s + (−0.0180 − 0.00585i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.178 + 0.983i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.178 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27287 - 1.52438i\)
\(L(\frac12)\) \(\approx\) \(1.27287 - 1.52438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.07 + 0.828i)T \)
11 \( 1 \)
good2 \( 1 + (0.471 + 0.649i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.67 + 0.544i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.563 - 0.182i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.05 + 1.45i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.15 + 5.72i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.706 - 2.17i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.49iT - 23T^{2} \)
29 \( 1 + (1.10 - 3.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.98 - 3.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.97 - 2.26i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.59 - 7.99i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.51iT - 43T^{2} \)
47 \( 1 + (1.83 - 0.596i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.40 - 1.92i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0118 + 0.0364i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.78 - 2.02i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.79iT - 67T^{2} \)
71 \( 1 + (-9.54 - 6.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.48 + 2.10i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.66 + 2.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.49 - 4.80i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (-3.15 - 4.34i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.15599946147051640761087961953, −9.557238388207038050391284538873, −8.883928781793084733644665119387, −7.981606028097498149459747890192, −6.93652317977873126823473751210, −5.67093505605948558314864198732, −5.09089309338126854277899165150, −3.12292144000243296393229519432, −2.31172436826159021395698816411, −1.20461422105080390205688595045, 2.08258192502213500112854929252, 3.10702590849033722613654411477, 4.03435661419623479016979287548, 5.68929455985179378425033803074, 6.48054628356275782868840903301, 7.63387090678457857767061824530, 8.162943226215385284505848010271, 9.315568103973938648514135231441, 9.513088604572924228623359688853, 10.74042870329993502389016324823

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