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Properties

Label 2-605-55.17-c1-0-10
Degree $2$
Conductor $605$
Sign $-0.282 - 0.959i$
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  + (0.435 + 0.221i)2-s + (1.12 − 0.178i)3-s + (−1.03 − 1.42i)4-s + (−1.14 + 1.92i)5-s + (0.529 + 0.171i)6-s + (−0.456 + 2.88i)7-s + (−0.287 − 1.81i)8-s + (−1.61 + 0.525i)9-s + (−0.922 + 0.583i)10-s + (−1.41 − 1.41i)12-s + (−1.32 + 2.60i)13-s + (−0.837 + 1.15i)14-s + (−0.940 + 2.36i)15-s + (−0.811 + 2.49i)16-s + (2.39 + 4.69i)17-s + (−0.820 − 0.130i)18-s + ⋯
L(s)  = 1  + (0.307 + 0.156i)2-s + (0.649 − 0.102i)3-s + (−0.517 − 0.712i)4-s + (−0.510 + 0.860i)5-s + (0.216 + 0.0701i)6-s + (−0.172 + 1.08i)7-s + (−0.101 − 0.641i)8-s + (−0.539 + 0.175i)9-s + (−0.291 + 0.184i)10-s + (−0.409 − 0.409i)12-s + (−0.368 + 0.723i)13-s + (−0.223 + 0.307i)14-s + (−0.242 + 0.611i)15-s + (−0.202 + 0.624i)16-s + (0.580 + 1.13i)17-s + (−0.193 − 0.0306i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.282 - 0.959i$
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: \((2,\ 605,\ (\ :1/2),\ -0.282 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724508 + 0.968843i\)
\(L(\frac12)\) \(\approx\) \(0.724508 + 0.968843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.14 - 1.92i)T \)
11 \( 1 \)
good2 \( 1 + (-0.435 - 0.221i)T + (1.17 + 1.61i)T^{2} \)
3 \( 1 + (-1.12 + 0.178i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (0.456 - 2.88i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (1.32 - 2.60i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-2.39 - 4.69i)T + (-9.99 + 13.7i)T^{2} \)
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 \( 1 + (1.14 + 0.181i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-2.08 + 2.87i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-5.07 - 5.07i)T + 43iT^{2} \)
47 \( 1 + (-0.575 - 3.63i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.31 - 0.670i)T + (31.1 + 42.8i)T^{2} \)
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 \( 1 + (14.6 + 2.31i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.71 - 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.95 + 3.03i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 + (4.57 - 8.97i)T + (-57.0 - 78.4i)T^{2} \)
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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

−10.89915481170493338485547049983, −9.863902465307796986419360934128, −9.221747525401927638480462672334, −8.277307037788234015178184984037, −7.46252828232219009930301356997, −6.10947407662814450110182783133, −5.68712426793760390315111548302, −4.23146843489574890506811709772, −3.23029290099596222859511299106, −2.04599684957305120377036694940, 0.56175705524728578125601297422, 2.89914625366935299535103887825, 3.60963948810782938858658343805, 4.61424088550816450178166348201, 5.42650095693347120587483795842, 7.26038689916755113217488971216, 7.76635365941391301024431437066, 8.686094808217239982701545493142, 9.282941810230008648464492700923, 10.30050483949521598248731396062

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