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Properties

Label 2-605-11.3-c1-0-10
Degree $2$
Conductor $605$
Sign $-0.0915 - 0.995i$
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  + (−2.12 + 1.54i)2-s + (0.414 + 1.27i)3-s + (1.50 − 4.64i)4-s + (−0.809 − 0.587i)5-s + (−2.84 − 2.06i)6-s + (−0.810 + 2.49i)7-s + (2.33 + 7.19i)8-s + (0.974 − 0.708i)9-s + 2.62·10-s + 6.54·12-s + (5.44 − 3.95i)13-s + (−2.12 − 6.54i)14-s + (0.414 − 1.27i)15-s + (−8.16 − 5.93i)16-s + (−1.44 − 1.04i)17-s + (−0.976 + 3.00i)18-s + ⋯
L(s)  = 1  + (−1.50 + 1.09i)2-s + (0.239 + 0.735i)3-s + (0.754 − 2.32i)4-s + (−0.361 − 0.262i)5-s + (−1.16 − 0.843i)6-s + (−0.306 + 0.943i)7-s + (0.826 + 2.54i)8-s + (0.324 − 0.236i)9-s + 0.829·10-s + 1.88·12-s + (1.50 − 1.09i)13-s + (−0.568 − 1.74i)14-s + (0.106 − 0.329i)15-s + (−2.04 − 1.48i)16-s + (−0.349 − 0.254i)17-s + (−0.230 + 0.708i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0915 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0915 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0915 - 0.995i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0915 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513450 + 0.562804i\)
\(L(\frac12)\) \(\approx\) \(0.513450 + 0.562804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (2.12 - 1.54i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.414 - 1.27i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.810 - 2.49i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.44 + 3.95i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.44 + 1.04i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.457 + 1.40i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.20T + 23T^{2} \)
29 \( 1 + (-0.363 + 1.11i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.69 + 2.68i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.210 - 0.646i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.69 - 5.23i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.80T + 43T^{2} \)
47 \( 1 + (-2.05 - 6.33i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.5 - 8.40i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.20 - 3.69i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.59 + 1.88i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.66T + 67T^{2} \)
71 \( 1 + (3.14 + 2.28i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.10 - 3.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-12.2 + 8.89i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.93 + 4.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.44T + 89T^{2} \)
97 \( 1 + (1.19 - 0.866i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.63083513356618082772275985834, −9.558195177637564729923040375368, −9.082772562474019039458283588087, −8.445363687392051105887466760031, −7.62540401279321306725937704752, −6.45930319509355345057612428121, −5.79080770071326693554334741225, −4.62700721074622616759673829491, −3.02080005528202729713475161226, −1.02175426202474875110354461175, 0.968743527746511937976207356399, 1.95696637516334821688838735459, 3.33663507691028467628997188261, 4.19719658992973968655131674548, 6.65528892460414299582169478387, 7.06056108245042753931694566621, 8.047753850892178441227452312175, 8.673867337396654789967456452525, 9.578690053228704299333921977957, 10.65073025244620318306767669989

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