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Properties

Label 2-605-55.4-c1-0-2
Degree $2$
Conductor $605$
Sign $-0.405 + 0.913i$
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  + (−0.972 + 1.33i)2-s + (1.87 + 0.610i)3-s + (−0.227 − 0.701i)4-s + (−2.02 + 0.938i)5-s + (−2.64 + 1.92i)6-s + (−2.13 + 0.693i)7-s + (−1.98 − 0.645i)8-s + (0.727 + 0.528i)9-s + (0.718 − 3.62i)10-s − 1.45i·12-s + (−2.17 + 2.99i)13-s + (1.14 − 3.52i)14-s + (−4.38 + 0.523i)15-s + (3.98 − 2.89i)16-s + (−1.30 − 1.79i)17-s + (−1.41 + 0.460i)18-s + ⋯
L(s)  = 1  + (−0.687 + 0.946i)2-s + (1.08 + 0.352i)3-s + (−0.113 − 0.350i)4-s + (−0.907 + 0.419i)5-s + (−1.07 + 0.784i)6-s + (−0.806 + 0.261i)7-s + (−0.702 − 0.228i)8-s + (0.242 + 0.176i)9-s + (0.227 − 1.14i)10-s − 0.420i·12-s + (−0.603 + 0.830i)13-s + (0.306 − 0.943i)14-s + (−1.13 + 0.135i)15-s + (0.997 − 0.724i)16-s + (−0.317 − 0.436i)17-s + (−0.333 + 0.108i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.405 + 0.913i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.405 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204947 - 0.315292i\)
\(L(\frac12)\) \(\approx\) \(0.204947 - 0.315292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.02 - 0.938i)T \)
11 \( 1 \)
good2 \( 1 + (0.972 - 1.33i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.87 - 0.610i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (2.13 - 0.693i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.17 - 2.99i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.30 + 1.79i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.63 + 5.02i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.85iT - 23T^{2} \)
29 \( 1 + (-0.0582 - 0.179i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.555 + 0.403i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.46 - 0.801i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.44 - 7.52i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.41iT - 43T^{2} \)
47 \( 1 + (11.4 + 3.71i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.43 - 10.2i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.106 - 0.326i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.40 + 1.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.650iT - 67T^{2} \)
71 \( 1 + (3.75 - 2.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.42 + 2.73i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.85 - 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.87 + 2.57i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (-1.33 + 1.83i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.27236578252072512258301565542, −9.657473416591815424267851189947, −9.446848009164045053539362512195, −8.558028679189580650794296772831, −7.79194484146671826279152863659, −7.02033703576673110822265784117, −6.32975438226872071004359252948, −4.69506783465890622425542669548, −3.38191198036659931187718235691, −2.79858619144594901166752905651, 0.21511469980823152279967284695, 1.85581466375305083219085759867, 3.08764341753369148398263535670, 3.69844509695983792587064065684, 5.34834379637723727969051247653, 6.74606681136497566980502613829, 7.892642548593722087091743244673, 8.367934437678347528667113938115, 9.200096929365784098734557845048, 10.03016547959964882790641167974

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