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Properties

Label 2-605-55.18-c1-0-2
Degree $2$
Conductor $605$
Sign $0.768 - 0.640i$
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  + (−0.380 − 2.40i)2-s + (0.532 − 0.271i)3-s + (−3.72 + 1.21i)4-s + (−0.622 − 2.14i)5-s + (−0.855 − 1.17i)6-s + (−1.17 + 2.30i)7-s + (2.11 + 4.15i)8-s + (−1.55 + 2.13i)9-s + (−4.92 + 2.31i)10-s + (−1.65 + 1.65i)12-s + (−2.77 + 0.439i)13-s + (5.97 + 1.94i)14-s + (−0.914 − 0.975i)15-s + (2.84 − 2.06i)16-s + (0.932 + 0.147i)17-s + (5.72 + 2.91i)18-s + ⋯
L(s)  = 1  + (−0.269 − 1.69i)2-s + (0.307 − 0.156i)3-s + (−1.86 + 0.605i)4-s + (−0.278 − 0.960i)5-s + (−0.349 − 0.480i)6-s + (−0.443 + 0.869i)7-s + (0.748 + 1.46i)8-s + (−0.517 + 0.712i)9-s + (−1.55 + 0.731i)10-s + (−0.478 + 0.478i)12-s + (−0.769 + 0.121i)13-s + (1.59 + 0.518i)14-s + (−0.236 − 0.251i)15-s + (0.710 − 0.516i)16-s + (0.226 + 0.0358i)17-s + (1.34 + 0.687i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.768 - 0.640i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.768 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117384 + 0.0424897i\)
\(L(\frac12)\) \(\approx\) \(0.117384 + 0.0424897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.622 + 2.14i)T \)
11 \( 1 \)
good2 \( 1 + (0.380 + 2.40i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (-0.532 + 0.271i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (1.17 - 2.30i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (2.77 - 0.439i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.932 - 0.147i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.28 - 3.94i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.104 - 0.104i)T + 23iT^{2} \)
29 \( 1 + (2.14 + 6.60i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.33 + 5.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.77 - 3.44i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (3.27 + 1.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.91 - 3.91i)T - 43iT^{2} \)
47 \( 1 + (-0.479 - 0.942i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (0.644 + 4.07i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (6.16 - 2.00i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.59 + 7.70i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.94 + 2.94i)T - 67iT^{2} \)
71 \( 1 + (1.02 - 0.744i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.57 + 0.804i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-3.51 - 2.55i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.29 - 8.18i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 4.23iT - 89T^{2} \)
97 \( 1 + (14.2 - 2.25i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.89647236949693466430519122857, −9.678469126290664279621013720584, −9.387045516205761565076047977895, −8.369021027667460028518534590602, −7.80544038615367905914521730540, −5.89258547596888755623808064179, −4.88152735993752257073320637222, −3.81148174316419752179760104704, −2.64247440105600960296766763517, −1.77584913056679898184491117561, 0.07009834972532837411703049205, 3.01819074345638089159645782917, 4.07222173164385900878029545287, 5.31490617706623701050427199715, 6.38624809187010808008595162654, 7.08013121567981357898194362189, 7.53775551697327506017767603549, 8.684741582229723776079494599062, 9.393240638982566612823765203454, 10.26621141409424501531719687588

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