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Properties

Label 2-605-55.17-c1-0-11
Degree $2$
Conductor $605$
Sign $0.996 - 0.0865i$
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  + (−1.61 + 0.256i)3-s + (−1.17 − 1.61i)4-s + (−1.93 + 1.11i)5-s + (−0.301 + 0.0978i)9-s + (2.31 + 2.31i)12-s + (2.85 − 2.29i)15-s + (−1.23 + 3.80i)16-s + (4.08 + 1.82i)20-s + (6.15 − 6.15i)23-s + (2.51 − 4.31i)25-s + (4.84 − 2.46i)27-s + (3.07 + 9.46i)31-s + (0.512 + 0.372i)36-s + (11.8 + 1.87i)37-s + (0.474 − 0.525i)45-s + ⋯
L(s)  = 1  + (−0.934 + 0.147i)3-s + (−0.587 − 0.809i)4-s + (−0.867 + 0.498i)5-s + (−0.100 + 0.0326i)9-s + (0.668 + 0.668i)12-s + (0.736 − 0.593i)15-s + (−0.309 + 0.951i)16-s + (0.912 + 0.408i)20-s + (1.28 − 1.28i)23-s + (0.503 − 0.863i)25-s + (0.931 − 0.474i)27-s + (0.552 + 1.69i)31-s + (0.0853 + 0.0620i)36-s + (1.94 + 0.308i)37-s + (0.0707 − 0.0782i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.996 - 0.0865i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.996 - 0.0865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637706 + 0.0276489i\)
\(L(\frac12)\) \(\approx\) \(0.637706 + 0.0276489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.256i)T + (2.85 - 0.927i)T^{2} \)
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 \( 1 + (-6.15 + 6.15i)T - 23iT^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.07 - 9.46i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.8 - 1.87i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-0.593 - 3.74i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (12.1 + 6.18i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (1.94 + 2.68i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \)
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 \( 1 + (-8.65 + 16.9i)T + (-57.0 - 78.4i)T^{2} \)
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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

−10.83511618663102963671574937381, −10.06179500900335590607276943985, −8.931508280354516691024276483090, −8.114217024910439502540795771622, −6.80419405877500369485194896023, −6.15479929537169981063476665965, −5.00566704361964299967054914084, −4.42356458298944261335026932412, −2.92224835235059757794198257516, −0.77312685715585986626464052787, 0.68332728283258264377283305363, 3.05023628175643515302836395474, 4.16791294111310432052190297062, 4.99470357843933588740673427421, 5.99105028933035575782249113164, 7.27823423649858112734214523508, 7.899476373314536078011297281587, 8.876916435284398101840736696094, 9.582853140393457379642416036900, 11.07887693729178029673176709341

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