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Properties

Label 2-605-55.9-c1-0-10
Degree $2$
Conductor $605$
Sign $0.605 - 0.795i$
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  + (−2.37 − 0.771i)2-s + (3.42 + 2.48i)4-s + (0.690 + 2.12i)5-s + (−1.81 + 2.49i)7-s + (−3.28 − 4.51i)8-s + (0.927 − 2.85i)9-s − 5.58i·10-s + (1.81 + 0.589i)13-s + (6.23 − 4.53i)14-s + (1.69 + 5.20i)16-s + (7.68 − 2.49i)17-s + (−4.40 + 6.06i)18-s + (−2.92 + 9.00i)20-s + (−4.04 + 2.93i)25-s + (−3.85 − 2.80i)26-s + ⋯
L(s)  = 1  + (−1.67 − 0.545i)2-s + (1.71 + 1.24i)4-s + (0.309 + 0.951i)5-s + (−0.685 + 0.943i)7-s + (−1.16 − 1.59i)8-s + (0.309 − 0.951i)9-s − 1.76i·10-s + (0.503 + 0.163i)13-s + (1.66 − 1.21i)14-s + (0.422 + 1.30i)16-s + (1.86 − 0.605i)17-s + (−1.03 + 1.42i)18-s + (−0.654 + 2.01i)20-s + (−0.809 + 0.587i)25-s + (−0.755 − 0.549i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.605 - 0.795i$
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: \((2,\ 605,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571682 + 0.283414i\)
\(L(\frac12)\) \(\approx\) \(0.571682 + 0.283414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.690 - 2.12i)T \)
11 \( 1 \)
good2 \( 1 + (2.37 + 0.771i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.81 - 2.49i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.81 - 0.589i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-7.68 + 2.49i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.76 - 8.50i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.23 - 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-2.47 - 7.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.99 - 9.62i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.693 - 0.225i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (78.4 + 57.0i)T^{2} \)
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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

−10.48370406100579209178332075167, −9.733010735113791925359779481368, −9.366502850087193700321926176908, −8.409804372390433964365642278693, −7.35857541212168444326861366705, −6.61942301417382924911296221811, −5.71951090970363319176993123780, −3.38790325635476916399042451375, −2.81586409174008933956114315885, −1.31575992675063519237466325425, 0.69598950160816027839286180676, 1.84633418876580622499644181005, 3.83042916928641970839337245654, 5.37962180283253116013557260706, 6.21060547324784790538673344755, 7.42392317233498346556413272602, 7.87152425163033124161182762482, 8.744430677692681505662278586247, 9.690707534982390686529316833838, 10.20095414339572668874212464557

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