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Properties

Label 2-605-55.49-c1-0-7
Degree $2$
Conductor $605$
Sign $-0.961 + 0.273i$
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  + (0.753 − 0.244i)2-s + (−1.48 + 2.04i)3-s + (−1.11 + 0.806i)4-s + (−1.12 + 1.93i)5-s + (−0.618 + 1.90i)6-s + (2.03 + 2.80i)7-s + (−1.57 + 2.16i)8-s + (−1.04 − 3.20i)9-s + (−0.372 + 1.73i)10-s − 3.46i·12-s + (2.22 + 1.61i)14-s + (−2.28 − 5.16i)15-s + (0.193 − 0.596i)16-s + (4.80 + 1.56i)17-s + (−1.57 − 2.16i)18-s + (−3.23 − 2.35i)19-s + ⋯
L(s)  = 1  + (0.532 − 0.173i)2-s + (−0.856 + 1.17i)3-s + (−0.555 + 0.403i)4-s + (−0.501 + 0.864i)5-s + (−0.252 + 0.776i)6-s + (0.769 + 1.05i)7-s + (−0.555 + 0.764i)8-s + (−0.347 − 1.06i)9-s + (−0.117 + 0.547i)10-s − 0.999i·12-s + (0.593 + 0.431i)14-s + (−0.589 − 1.33i)15-s + (0.0484 − 0.149i)16-s + (1.16 + 0.378i)17-s + (−0.370 − 0.509i)18-s + (−0.742 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.961 + 0.273i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128060 - 0.916909i\)
\(L(\frac12)\) \(\approx\) \(0.128060 - 0.916909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.12 - 1.93i)T \)
11 \( 1 \)
good2 \( 1 + (-0.753 + 0.244i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.48 - 2.04i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-2.03 - 2.80i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.80 - 1.56i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.52iT - 23T^{2} \)
29 \( 1 + (-2.22 + 1.61i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.733 + 2.25i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.48 - 8.92i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.22 + 1.61i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (-3.89 + 5.36i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.01 - 0.979i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.31 - 0.956i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.32 - 10.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.644iT - 67T^{2} \)
71 \( 1 + (-2.19 + 6.76i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.07 - 5.60i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.93 - 12.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (6.30 + 2.04i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + (-3.90 + 1.26i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.36730926981534697030982136739, −10.40914508707435381040959449706, −9.572904362555149753325859121335, −8.529088352479536430244052752101, −7.76823302027808135636684917665, −6.19576027906529259766896978178, −5.40195950664428299219039843652, −4.60642836333675170536756040106, −3.76955860659752155253116475589, −2.67222675711988413606763834955, 0.54086608481602934859280654824, 1.41622079260975717195869397326, 3.84171159527231735529504412375, 4.73627142364151567664038041630, 5.51262124133148122665486021098, 6.43951242817080010782523670440, 7.48794044896763799054159969213, 8.078762634854383808951647986982, 9.237136879699758406116537967370, 10.40832997403569806391028668760

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