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Properties

Label 2-605-11.3-c1-0-8
Degree $2$
Conductor $605$
Sign $0.138 - 0.990i$
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  + (−1.40 + 1.01i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (1.40 + 1.01i)6-s + (−0.535 + 1.64i)7-s + (−0.535 − 1.64i)8-s + (1.61 − 1.17i)9-s − 1.73·10-s − 0.999·12-s + (−2.80 + 2.03i)13-s + (−0.927 − 2.85i)14-s + (0.309 − 0.951i)15-s + (4.04 + 2.93i)16-s + (5.60 + 4.07i)17-s + (−1.07 + 3.29i)18-s + ⋯
L(s)  = 1  + (−0.990 + 0.719i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.572 + 0.415i)6-s + (−0.202 + 0.622i)7-s + (−0.189 − 0.582i)8-s + (0.539 − 0.391i)9-s − 0.547·10-s − 0.288·12-s + (−0.777 + 0.564i)13-s + (−0.247 − 0.762i)14-s + (0.0797 − 0.245i)15-s + (1.01 + 0.734i)16-s + (1.35 + 0.987i)17-s + (−0.252 + 0.776i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.609259 + 0.529851i\)
\(L(\frac12)\) \(\approx\) \(0.609259 + 0.529851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (1.40 - 1.01i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.535 - 1.64i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.80 - 2.03i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.60 - 4.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.07 + 3.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.47 + 4.70i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.47 - 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.74 - 11.5i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.66T + 43T^{2} \)
47 \( 1 + (-2.78 - 8.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.85 - 3.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.70 - 11.4i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.00 + 5.09i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + (-9.70 - 7.05i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.40 + 6.10i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.80 + 2.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-8.09 + 5.87i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.49636476716077315907894199955, −9.654701408340186037396432074286, −9.207314257121714687020679730504, −8.015702279328622753681912981577, −7.44232931310777917637515276753, −6.40434701715312393673356329058, −6.01723694850754941447023122607, −4.38197635984942387694160960670, −2.84565272007799418384635702687, −1.23694867508259285163514227712, 0.75619212373321242694797397195, 2.19457092929033037036542816359, 3.56361716323045998510796647963, 4.93396996950812298034238074401, 5.64040198056616828443335434572, 7.25962983676676679942671270092, 7.926370578936871790652784835756, 9.114143492072618353226735095796, 9.792438169684518254547302586090, 10.36594652890299019596130953000

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