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Properties

Label 2-605-55.14-c1-0-20
Degree $2$
Conductor $605$
Sign $0.888 + 0.458i$
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  + (1.18 + 1.63i)2-s + (−2.77 + 0.900i)3-s + (−0.646 + 1.98i)4-s + (−2.08 + 0.818i)5-s + (−4.76 − 3.46i)6-s + (−3.05 − 0.992i)7-s + (−0.176 + 0.0573i)8-s + (4.44 − 3.23i)9-s + (−3.81 − 2.43i)10-s − 6.09i·12-s + (0.381 + 0.524i)13-s + (−2.00 − 6.17i)14-s + (5.03 − 4.14i)15-s + (3.08 + 2.23i)16-s + (−0.698 + 0.961i)17-s + (10.5 + 3.43i)18-s + ⋯
L(s)  = 1  + (0.840 + 1.15i)2-s + (−1.60 + 0.520i)3-s + (−0.323 + 0.994i)4-s + (−0.930 + 0.366i)5-s + (−1.94 − 1.41i)6-s + (−1.15 − 0.375i)7-s + (−0.0624 + 0.0202i)8-s + (1.48 − 1.07i)9-s + (−1.20 − 0.768i)10-s − 1.76i·12-s + (0.105 + 0.145i)13-s + (−0.536 − 1.65i)14-s + (1.29 − 1.07i)15-s + (0.770 + 0.559i)16-s + (−0.169 + 0.233i)17-s + (2.49 + 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.313766 - 0.0761014i\)
\(L(\frac12)\) \(\approx\) \(0.313766 - 0.0761014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 - 0.818i)T \)
11 \( 1 \)
good2 \( 1 + (-1.18 - 1.63i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.77 - 0.900i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (3.05 + 0.992i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.381 - 0.524i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.698 - 0.961i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.584 + 1.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + (-1.28 + 3.96i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.39 - 4.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.95 + 0.636i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.02 + 3.14i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + (2.77 - 0.900i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.77 + 5.19i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.14 + 3.53i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.29 - 4.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.37iT - 67T^{2} \)
71 \( 1 + (12.7 + 9.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.43 - 1.76i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.3 - 8.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.25 + 8.60i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (8.89 + 12.2i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.62007408364461266338858515145, −10.09426882270136296410206532416, −8.648939991287746191956225722748, −7.26210953877370340709055104125, −6.75228853589877047237453633984, −6.12992671508259370001427191304, −5.15046453235004334632130281772, −4.27213036278442188510725898434, −3.54918466529557324658917995025, −0.18049482553238592068130943223, 1.34225857008438633074450437754, 3.07062706793518335849269873954, 4.10606271831000510384596941838, 5.11462067041561130406588104697, 5.87402782923080880270724826898, 6.90357988979132935010691381776, 7.84788121065725077421507947618, 9.364619369046712887566099457936, 10.32550347505312256393377964149, 11.20540661415039795834485011601

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