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Properties

Label 2-605-5.4-c1-0-8
Degree $2$
Conductor $605$
Sign $-0.537 + 0.843i$
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  + 2.02i·2-s + 2.91i·3-s − 2.09·4-s + (1.20 − 1.88i)5-s − 5.89·6-s + 3.21i·7-s − 0.185i·8-s − 5.49·9-s + (3.81 + 2.43i)10-s − 6.09i·12-s + 0.648i·13-s − 6.49·14-s + (5.49 + 3.50i)15-s − 3.80·16-s + 1.18i·17-s − 11.1i·18-s + ⋯
L(s)  = 1  + 1.43i·2-s + 1.68i·3-s − 1.04·4-s + (0.537 − 0.843i)5-s − 2.40·6-s + 1.21i·7-s − 0.0656i·8-s − 1.83·9-s + (1.20 + 0.768i)10-s − 1.76i·12-s + 0.179i·13-s − 1.73·14-s + (1.41 + 0.904i)15-s − 0.952·16-s + 0.288i·17-s − 2.62i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.537 + 0.843i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679039 - 1.23818i\)
\(L(\frac12)\) \(\approx\) \(0.679039 - 1.23818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.20 + 1.88i)T \)
11 \( 1 \)
good2 \( 1 - 2.02iT - 2T^{2} \)
3 \( 1 - 2.91iT - 3T^{2} \)
7 \( 1 - 3.21iT - 7T^{2} \)
13 \( 1 - 0.648iT - 13T^{2} \)
17 \( 1 - 1.18iT - 17T^{2} \)
19 \( 1 - 1.89T + 19T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 - 7.89T + 31T^{2} \)
37 \( 1 + 2.05iT - 37T^{2} \)
41 \( 1 - 3.30T + 41T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 - 2.91iT - 47T^{2} \)
53 \( 1 - 6.41iT - 53T^{2} \)
59 \( 1 - 3.71T + 59T^{2} \)
61 \( 1 - 7.78T + 61T^{2} \)
67 \( 1 + 2.37iT - 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + 5.71iT - 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−11.07556258991563424862151220918, −9.914016056813349404541281982692, −9.256459190459758596699980461706, −8.709822353490290453692806268781, −7.996384529183988801003161040089, −6.33918366026832021436167432826, −5.68588706101843741551877762544, −4.99281325790098111306086947505, −4.27634006971390866249887173090, −2.58991873612344909437720060403, 0.799726785967387345463535131481, 1.84783548590967637207233878773, 2.82079919852587197901969532535, 3.82418122828260801505481265581, 5.57902069429718558505282584899, 6.89522543849059094887561167652, 7.10592753464981190309650322724, 8.233593549221042021639156092644, 9.616608340924031448274696396590, 10.25889696365562216482170600117

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