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Properties

Label 2-13e2-13.4-c1-0-1
Degree $2$
Conductor $169$
Sign $0.802 + 0.596i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
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  + (−1.94 − 1.12i)2-s + (0.277 − 0.480i)3-s + (1.52 + 2.64i)4-s + 1.44i·5-s + (−1.07 + 0.623i)6-s + (1.77 − 1.02i)7-s − 2.35i·8-s + (1.34 + 2.33i)9-s + (1.62 − 2.81i)10-s + (2.21 + 1.27i)11-s + 1.69·12-s − 4.60·14-s + (0.694 + 0.400i)15-s + (0.400 − 0.694i)16-s + (−2.64 − 4.58i)17-s − 6.04i·18-s + ⋯
L(s)  = 1  + (−1.37 − 0.794i)2-s + (0.160 − 0.277i)3-s + (0.762 + 1.32i)4-s + 0.646i·5-s + (−0.440 + 0.254i)6-s + (0.670 − 0.387i)7-s − 0.833i·8-s + (0.448 + 0.777i)9-s + (0.513 − 0.889i)10-s + (0.667 + 0.385i)11-s + 0.488·12-s − 1.23·14-s + (0.179 + 0.103i)15-s + (0.100 − 0.173i)16-s + (−0.642 − 1.11i)17-s − 1.42i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.802 + 0.596i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.802 + 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668351 - 0.221014i\)
\(L(\frac12)\) \(\approx\) \(0.668351 - 0.221014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (1.94 + 1.12i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.277 + 0.480i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.44iT - 5T^{2} \)
7 \( 1 + (-1.77 + 1.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.21 - 1.27i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.64 + 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.945 - 1.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.26iT - 31T^{2} \)
37 \( 1 + (4.63 + 2.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.10 - 0.637i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.06 - 5.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.95iT - 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 + (10.5 - 6.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.28 + 7.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.499 - 0.288i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.97 + 2.29i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 7.72iT - 83T^{2} \)
89 \( 1 + (5.72 + 3.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 - 5.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.26578241491693098621397411334, −11.27973318538415097704269446841, −10.74692916496189393176385416905, −9.693931528745197774175762994917, −8.808424785226795388093184921242, −7.52256125941664867267212937763, −7.08822014038857722589233254951, −4.83755151439915759275069797506, −2.89318351888349715707419521287, −1.50863080964455333884930204928, 1.33021498981846409212911426297, 4.03339081718526163649854059688, 5.69484761927557375310077556311, 6.78588052432816725765530899507, 8.059141811987483046662798755836, 8.784788236147696033282010979003, 9.471369280541764344479069089584, 10.49426824377167080702488870945, 11.73476115597629617808007338543, 12.75202203300852380347522503127

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