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Properties

Label 2-605-55.4-c1-0-43
Degree $2$
Conductor $605$
Sign $-0.858 - 0.513i$
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  + (1.48 − 2.04i)2-s + (−0.753 − 0.244i)3-s + (−1.35 − 4.15i)4-s + (−1.80 − 1.31i)5-s + (−1.61 + 1.17i)6-s + (3.29 − 1.07i)7-s + (−5.69 − 1.85i)8-s + (−1.91 − 1.39i)9-s + (−5.37 + 1.73i)10-s + 3.46i·12-s + (2.70 − 8.31i)14-s + (1.03 + 1.43i)15-s + (−5.15 + 3.74i)16-s + (0.931 + 1.28i)17-s + (−5.69 + 1.85i)18-s + (−1.23 + 3.80i)19-s + ⋯
L(s)  = 1  + (1.04 − 1.44i)2-s + (−0.435 − 0.141i)3-s + (−0.675 − 2.07i)4-s + (−0.807 − 0.589i)5-s + (−0.660 + 0.479i)6-s + (1.24 − 0.404i)7-s + (−2.01 − 0.654i)8-s + (−0.639 − 0.464i)9-s + (−1.69 + 0.547i)10-s + 0.999i·12-s + (0.722 − 2.22i)14-s + (0.268 + 0.370i)15-s + (−1.28 + 0.936i)16-s + (0.225 + 0.310i)17-s + (−1.34 + 0.436i)18-s + (−0.283 + 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.858 - 0.513i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461369 + 1.67005i\)
\(L(\frac12)\) \(\approx\) \(0.461369 + 1.67005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.80 + 1.31i)T \)
11 \( 1 \)
good2 \( 1 + (-1.48 + 2.04i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.753 + 0.244i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-3.29 + 1.07i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.931 - 1.28i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 - 3.80i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.792iT - 23T^{2} \)
29 \( 1 + (2.70 + 8.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.72 + 1.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.03 - 0.335i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.70 + 8.31i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (-6.30 - 2.04i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.93 + 8.16i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.27 - 7.01i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.602 + 0.437i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.30iT - 67T^{2} \)
71 \( 1 + (-8.18 + 5.94i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.58 + 2.14i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.01 - 0.737i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.89 + 5.36i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + (3.43 - 4.72i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.73910767057184856477538081207, −9.573891805690852359643949011231, −8.478169461053225166123907556615, −7.57830855722491542602952338131, −5.94428991585294720242660165249, −5.21305984850961839076617805261, −4.24484374470283367171421623220, −3.60937310557964884744266094379, −1.99940728107360899920970340015, −0.74939866285875924970492659256, 2.76413225957731913798558968228, 4.05480792705785020417001558250, 4.99917258998100312043390401042, 5.49298430190180678097670844447, 6.68082458140520145197528176649, 7.41766143152855810292328342534, 8.210524776197548948169320283643, 8.838905883823067803856793889060, 10.70101543613684959554167855014, 11.32837387427112147371624596665

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