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Properties

Label 2-605-55.28-c1-0-15
Degree $2$
Conductor $605$
Sign $-0.522 - 0.852i$
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  + (0.477 + 3.01i)3-s + (1.17 + 1.61i)4-s + (1.93 − 1.11i)5-s + (−6.00 + 1.95i)9-s + (−4.31 + 4.31i)12-s + (4.28 + 5.31i)15-s + (−1.23 + 3.80i)16-s + (4.08 + 1.82i)20-s + (2.84 + 2.84i)23-s + (2.51 − 4.31i)25-s + (−4.59 − 9.01i)27-s + (−3.07 − 9.46i)31-s + (−10.2 − 7.42i)36-s + (−0.326 + 2.06i)37-s + (−9.47 + 10.4i)45-s + ⋯
L(s)  = 1  + (0.275 + 1.74i)3-s + (0.587 + 0.809i)4-s + (0.867 − 0.498i)5-s + (−2.00 + 0.650i)9-s + (−1.24 + 1.24i)12-s + (1.10 + 1.37i)15-s + (−0.309 + 0.951i)16-s + (0.912 + 0.408i)20-s + (0.592 + 0.592i)23-s + (0.503 − 0.863i)25-s + (−0.884 − 1.73i)27-s + (−0.552 − 1.69i)31-s + (−1.70 − 1.23i)36-s + (−0.0536 + 0.338i)37-s + (−1.41 + 1.56i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982079 + 1.75471i\)
\(L(\frac12)\) \(\approx\) \(0.982079 + 1.75471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.93 + 1.11i)T \)
11 \( 1 \)
good2 \( 1 + (-1.17 - 1.61i)T^{2} \)
3 \( 1 + (-0.477 - 3.01i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.84 - 2.84i)T + 23iT^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.07 + 9.46i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.326 - 2.06i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-13.0 + 2.06i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (2.33 - 4.57i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (1.94 + 2.68i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.4 + 11.4i)T - 67iT^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (4.44 + 2.26i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.86324586219595503306931432985, −9.971908091010917035501488952535, −9.274941345793095065341840196195, −8.625242359272173083449570882304, −7.61642207563642311726616739208, −6.21340697785639706190245631558, −5.28021108176759087878200175376, −4.30591990323105038689797030627, −3.37391074223089295939623336695, −2.28061235787351031197086339530, 1.15639434212683810015343041122, 2.12153348287461976380525188887, 2.97324635191208130325922496414, 5.27372087651876170515540448400, 6.09344605917648284300985970146, 6.83188009846513794516445057789, 7.29472530899740703333946769985, 8.529561192827789210893133296102, 9.434457288526197697714494039911, 10.54198138188945701665731043668

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