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Properties

Label 2-605-55.49-c1-0-29
Degree $2$
Conductor $605$
Sign $0.215 + 0.976i$
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  + (−1.57 + 0.511i)2-s + (−1.16 + 1.59i)3-s + (0.596 − 0.433i)4-s + (0.264 − 2.22i)5-s + (1.00 − 3.10i)6-s + (1.31 + 1.81i)7-s + (1.22 − 1.68i)8-s + (−0.278 − 0.855i)9-s + (0.718 + 3.62i)10-s + 1.45i·12-s + (−3.51 + 1.14i)13-s + (−3.00 − 2.18i)14-s + (3.23 + 3.00i)15-s + (−1.52 + 4.68i)16-s + (−2.11 − 0.687i)17-s + (0.875 + 1.20i)18-s + ⋯
L(s)  = 1  + (−1.11 + 0.361i)2-s + (−0.670 + 0.922i)3-s + (0.298 − 0.216i)4-s + (0.118 − 0.992i)5-s + (0.412 − 1.26i)6-s + (0.498 + 0.685i)7-s + (0.434 − 0.597i)8-s + (−0.0926 − 0.285i)9-s + (0.227 + 1.14i)10-s + 0.420i·12-s + (−0.975 + 0.317i)13-s + (−0.802 − 0.583i)14-s + (0.836 + 0.774i)15-s + (−0.380 + 1.17i)16-s + (−0.513 − 0.166i)17-s + (0.206 + 0.283i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.215 + 0.976i$
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: \((2,\ 605,\ (\ :1/2),\ 0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153053 - 0.122939i\)
\(L(\frac12)\) \(\approx\) \(0.153053 - 0.122939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.264 + 2.22i)T \)
11 \( 1 \)
good2 \( 1 + (1.57 - 0.511i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.16 - 1.59i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.31 - 1.81i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (3.51 - 1.14i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.11 + 0.687i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.27 + 3.10i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.85iT - 23T^{2} \)
29 \( 1 + (0.152 - 0.110i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.212 - 0.653i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.52 - 2.09i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.40 - 4.65i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.41iT - 43T^{2} \)
47 \( 1 + (-7.06 + 9.71i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (12.0 - 3.91i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.278 - 0.202i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.535 - 1.64i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.650iT - 67T^{2} \)
71 \( 1 + (-1.43 + 4.42i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.20 + 7.16i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.23 + 6.88i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.02 + 0.983i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (-2.15 + 0.700i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.26003401311981350847819994978, −9.432938495213556849499248770437, −8.864818698328639370377621489710, −8.161194299417006715385739916577, −7.05150156361634206827375106826, −5.84443444050506356758004605509, −4.74080023690946515457931634283, −4.37815725098903402443011831901, −2.08212313841406821085477471350, −0.17967563185583109491568685734, 1.36146362287046161308518092794, 2.46793321726129781113476644827, 4.23548916492204335405102910086, 5.64127563383909497991131822841, 6.61471994640120443375679876844, 7.55470435570921782385662138890, 7.88353675432072300936197081596, 9.310088455494365781808808415673, 10.05791583873264037143338789055, 10.96341450742596929723251430083

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