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Properties

Label 2-605-11.4-c1-0-2
Degree $2$
Conductor $605$
Sign $-0.394 - 0.918i$
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  + (−1.12 − 0.820i)2-s + (−0.511 + 1.57i)3-s + (−0.0158 − 0.0488i)4-s + (0.809 − 0.587i)5-s + (1.86 − 1.35i)6-s + (1.45 + 4.47i)7-s + (−0.884 + 2.72i)8-s + (0.209 + 0.152i)9-s − 1.39·10-s + 0.0850·12-s + (−4.08 − 2.96i)13-s + (2.03 − 6.24i)14-s + (0.511 + 1.57i)15-s + (3.15 − 2.28i)16-s + (−4.29 + 3.12i)17-s + (−0.111 − 0.344i)18-s + ⋯
L(s)  = 1  + (−0.798 − 0.580i)2-s + (−0.295 + 0.908i)3-s + (−0.00793 − 0.0244i)4-s + (0.361 − 0.262i)5-s + (0.763 − 0.554i)6-s + (0.549 + 1.69i)7-s + (−0.312 + 0.962i)8-s + (0.0699 + 0.0508i)9-s − 0.441·10-s + 0.0245·12-s + (−1.13 − 0.823i)13-s + (0.542 − 1.67i)14-s + (0.132 + 0.406i)15-s + (0.787 − 0.572i)16-s + (−1.04 + 0.757i)17-s + (−0.0263 − 0.0812i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.394 - 0.918i$
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: \((2,\ 605,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326824 + 0.496067i\)
\(L(\frac12)\) \(\approx\) \(0.326824 + 0.496067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 + 0.820i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.511 - 1.57i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.45 - 4.47i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.08 + 2.96i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.29 - 3.12i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.698 + 2.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + (-0.862 - 2.65i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.02 + 2.19i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.244 + 0.753i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.90 - 5.85i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.70T + 43T^{2} \)
47 \( 1 + (3.47 - 10.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.850 + 0.617i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.40 - 4.31i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.99 - 5.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (-0.850 + 0.617i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.12 + 9.60i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.61 - 5.53i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.55 + 1.85i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + (-14.3 - 10.4i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.82517190584458375960221696809, −9.973573411005704368452172690994, −9.304056223623771280550965478540, −8.748756946331425570250204523889, −7.81481230655793803387046943932, −6.09840151141328069512086249444, −5.20355922216804602081959309160, −4.75570451022214380492007964087, −2.77244574812115674029136040029, −1.82993145700979603353062405621, 0.43714508545519804477771454640, 1.86006962058139677565405126063, 3.75571115464533009993284224340, 4.81511532207382229292992014713, 6.47264172484908533981350114356, 7.14249185461797288182991644487, 7.32784816373216485333624726374, 8.394498350244491680628369672800, 9.552020239442117113993496097326, 10.12186798672027512148835089683

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