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Properties

Label 2-605-55.28-c1-0-22
Degree $2$
Conductor $605$
Sign $0.369 - 0.929i$
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  + (0.221 − 0.435i)2-s + (0.178 + 1.12i)3-s + (1.03 + 1.42i)4-s + (−0.207 + 2.22i)5-s + (0.529 + 0.171i)6-s + (2.88 + 0.456i)7-s + (1.81 − 0.287i)8-s + (1.61 − 0.525i)9-s + (0.922 + 0.583i)10-s + (−1.41 + 1.41i)12-s + (−2.60 − 1.32i)13-s + (0.837 − 1.15i)14-s + (−2.54 + 0.163i)15-s + (−0.811 + 2.49i)16-s + (4.69 − 2.39i)17-s + (0.130 − 0.820i)18-s + ⋯
L(s)  = 1  + (0.156 − 0.307i)2-s + (0.102 + 0.649i)3-s + (0.517 + 0.712i)4-s + (−0.0928 + 0.995i)5-s + (0.216 + 0.0701i)6-s + (1.08 + 0.172i)7-s + (0.641 − 0.101i)8-s + (0.539 − 0.175i)9-s + (0.291 + 0.184i)10-s + (−0.409 + 0.409i)12-s + (−0.723 − 0.368i)13-s + (0.223 − 0.307i)14-s + (−0.656 + 0.0421i)15-s + (−0.202 + 0.624i)16-s + (1.13 − 0.580i)17-s + (0.0306 − 0.193i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.369 - 0.929i$
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: \((2,\ 605,\ (\ :1/2),\ 0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76212 + 1.19608i\)
\(L(\frac12)\) \(\approx\) \(1.76212 + 1.19608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.207 - 2.22i)T \)
11 \( 1 \)
good2 \( 1 + (-0.221 + 0.435i)T + (-1.17 - 1.61i)T^{2} \)
3 \( 1 + (-0.178 - 1.12i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-2.88 - 0.456i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (2.60 + 1.32i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-4.69 + 2.39i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.31 + 2.40i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.12 + 2.12i)T + 23iT^{2} \)
29 \( 1 + (2.13 - 1.55i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.08 + 6.41i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.181 - 1.14i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.08 + 2.87i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (5.07 - 5.07i)T - 43iT^{2} \)
47 \( 1 + (-3.63 + 0.575i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.670 - 1.31i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-0.943 - 1.29i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.59 + 2.14i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.31 + 1.31i)T - 67iT^{2} \)
71 \( 1 + (-0.887 + 2.73i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.31 + 14.6i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (4.71 + 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.03 - 5.95i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (-8.97 - 4.57i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.78317420035256686550586198572, −10.24965748791995917112696056704, −9.219397588151871205899365894864, −7.87237091868162941952583132635, −7.51446031060541944119480109078, −6.43464217015234398952417832474, −5.01575923945172470784224577336, −4.08085108784898121710912475112, −3.09096694296901263680913989821, −2.04206435871462292856725186575, 1.33037000539403003184679811542, 1.94827147996801368972601564351, 4.16276826562887051045916347798, 5.04884096336870715856874523032, 5.84426749493614586198422316052, 7.05531428539953234166260278309, 7.75525354113225478256548821667, 8.421502341151827739319839826605, 9.740309132279210997733865178101, 10.42467671396801891479723769342

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