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Properties

Label 2-605-11.4-c1-0-5
Degree $2$
Conductor $605$
Sign $-0.138 - 0.990i$
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  + (0.389 + 0.282i)2-s + (−0.494 + 1.52i)3-s + (−0.546 − 1.68i)4-s + (−0.809 + 0.587i)5-s + (−0.623 + 0.452i)6-s + (0.148 + 0.457i)7-s + (0.560 − 1.72i)8-s + (0.351 + 0.255i)9-s − 0.480·10-s + 2.83·12-s + (3.87 + 2.81i)13-s + (−0.0714 + 0.219i)14-s + (−0.494 − 1.52i)15-s + (−2.15 + 1.56i)16-s + (−2.02 + 1.47i)17-s + (0.0645 + 0.198i)18-s + ⋯
L(s)  = 1  + (0.275 + 0.199i)2-s + (−0.285 + 0.879i)3-s + (−0.273 − 0.841i)4-s + (−0.361 + 0.262i)5-s + (−0.254 + 0.184i)6-s + (0.0561 + 0.172i)7-s + (0.198 − 0.609i)8-s + (0.117 + 0.0851i)9-s − 0.152·10-s + 0.817·12-s + (1.07 + 0.781i)13-s + (−0.0190 + 0.0587i)14-s + (−0.127 − 0.393i)15-s + (−0.539 + 0.391i)16-s + (−0.490 + 0.356i)17-s + (0.0152 + 0.0468i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841144 + 0.967206i\)
\(L(\frac12)\) \(\approx\) \(0.841144 + 0.967206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.389 - 0.282i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.494 - 1.52i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.148 - 0.457i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.87 - 2.81i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.02 - 1.47i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.77 - 5.47i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 + (-2.78 - 8.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.44 + 4.68i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.60 - 4.94i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.61 - 8.03i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.53T + 43T^{2} \)
47 \( 1 + (-2.96 + 9.13i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.93 + 3.58i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.855 - 2.63i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.25 - 2.36i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.60T + 67T^{2} \)
71 \( 1 + (-2.23 + 1.62i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.54 + 4.75i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.93 - 2.13i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.38 + 1.00i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-2.94 - 2.13i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.83415917072367724049309924882, −10.16534306679870201237317478676, −9.251647406794227495817275379402, −8.484917275930398455315061134110, −7.09214530123045348332779156975, −6.20911619871374898840894237649, −5.30986656177536347196562900106, −4.36433053116600039510597649975, −3.66142170066866460852963562564, −1.61892871296965236944968673278, 0.73286493981476728024790296377, 2.49215683949568160297644730322, 3.74899707601361347557049055793, 4.65302971374014379779748933630, 5.90197180694422331854757457578, 7.04601977552391143856214675798, 7.61651991777951553421430467580, 8.594511194806708470310937447140, 9.269347760042271815597052489633, 10.91270526817797538607733420219

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