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Properties

Label 2-605-55.13-c1-0-4
Degree $2$
Conductor $605$
Sign $-0.901 - 0.432i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.901 - 0.432i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136103 + 0.597882i\)
\(L(\frac12)\) \(\approx\) \(0.136103 + 0.597882i\)
\(L(\frac{3}{2})\) 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

−11.15312641133824311988399316982, −9.739102057940438225701604114324, −8.833597262737004716943905286026, −8.541852769806908742618249727164, −8.070215932490612166155854501596, −6.64767857536874909516613812304, −5.47579793726587722873198080724, −4.26180786787002619125305221766, −3.53898198378716303231887759319, −2.04144337949138790508586522627, 0.33700294685502737548680926770, 2.21828861903237745059440999180, 3.45201321817247918035484323378, 4.49066086622539507817100987135, 5.70832097633711582231159955254, 6.90400012321687091051042808480, 7.75466026752053955728762754773, 8.502970431526593091118652196543, 9.446830892073407767359141171975, 10.33756551791993920359395463376

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