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Properties

Label 2-605-55.49-c1-0-26
Degree $2$
Conductor $605$
Sign $0.774 + 0.632i$
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  + (−2.48 + 0.805i)2-s + (−1.25 + 1.73i)3-s + (3.88 − 2.82i)4-s + (0.0494 + 2.23i)5-s + (1.72 − 5.30i)6-s + (−0.581 − 0.800i)7-s + (−4.29 + 5.90i)8-s + (−0.487 − 1.50i)9-s + (−1.92 − 5.50i)10-s + 10.2i·12-s + (2.92 − 0.950i)13-s + (2.08 + 1.51i)14-s + (−3.93 − 2.72i)15-s + (2.91 − 8.98i)16-s + (−1.53 − 0.500i)17-s + (2.41 + 3.33i)18-s + ⋯
L(s)  = 1  + (−1.75 + 0.569i)2-s + (−0.726 + 0.999i)3-s + (1.94 − 1.41i)4-s + (0.0220 + 0.999i)5-s + (0.703 − 2.16i)6-s + (−0.219 − 0.302i)7-s + (−1.51 + 2.08i)8-s + (−0.162 − 0.500i)9-s + (−0.608 − 1.74i)10-s + 2.96i·12-s + (0.811 − 0.263i)13-s + (0.557 + 0.405i)14-s + (−1.01 − 0.703i)15-s + (0.729 − 2.24i)16-s + (−0.373 − 0.121i)17-s + (0.570 + 0.785i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126939 - 0.0452374i\)
\(L(\frac12)\) \(\approx\) \(0.126939 - 0.0452374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0494 - 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (2.48 - 0.805i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.25 - 1.73i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (0.581 + 0.800i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2.92 + 0.950i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.53 + 0.500i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.32 + 3.86i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 + (5.91 - 4.29i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.816 + 2.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.64 + 2.26i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.12 - 0.814i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (1.25 - 1.73i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.26 - 0.734i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.90 + 7.19i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.309 + 0.953i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + (0.0753 - 0.231i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.03 - 11.0i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.37 - 7.31i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.66 - 1.84i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (12.1 - 3.93i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.53674069493804335500934570196, −9.769452460871049861227031200613, −8.956782575072992674882470137302, −8.039596414681602983504561830886, −6.91697461964601668948056967389, −6.43067119166153440646311010305, −5.41259998402420080394161458569, −3.90433608003938784845617705836, −2.26186301030696848610749573242, −0.15524006763822526641901806599, 1.24013515268328811472047628715, 2.02033336616267543309720829673, 3.83870661077717029091887891677, 5.73226524989209244676931648483, 6.47553183067494371034720778088, 7.50262472744109749979151769296, 8.271536167787379881604422650830, 8.993605862392345656830982951129, 9.723844365553042404968782108963, 10.78839663370404837388134752567

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