Properties

Label 2-13e2-169.10-c1-0-0
Degree $2$
Conductor $169$
Sign $-0.986 - 0.161i$
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  + (0.164 + 0.0336i)2-s + (−1.12 + 0.182i)3-s + (−1.81 − 0.772i)4-s + (−0.123 + 0.501i)5-s + (−0.191 − 0.00772i)6-s + (−2.88 + 1.37i)7-s + (−0.549 − 0.379i)8-s + (−1.61 + 0.538i)9-s + (−0.0372 + 0.0784i)10-s + (0.240 − 0.719i)11-s + (2.18 + 0.537i)12-s + (−3.58 − 0.358i)13-s + (−0.522 + 0.128i)14-s + (0.0473 − 0.586i)15-s + (2.65 + 2.76i)16-s + (0.268 + 0.565i)17-s + ⋯
L(s)  = 1  + (0.116 + 0.0237i)2-s + (−0.649 + 0.105i)3-s + (−0.906 − 0.386i)4-s + (−0.0552 + 0.224i)5-s + (−0.0782 − 0.00315i)6-s + (−1.09 + 0.518i)7-s + (−0.194 − 0.134i)8-s + (−0.537 + 0.179i)9-s + (−0.0117 + 0.0248i)10-s + (0.0723 − 0.216i)11-s + (0.629 + 0.155i)12-s + (−0.995 − 0.0993i)13-s + (−0.139 + 0.0344i)14-s + (0.0122 − 0.151i)15-s + (0.663 + 0.690i)16-s + (0.0650 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00745458 + 0.0919399i\)
\(L(\frac12)\) \(\approx\) \(0.00745458 + 0.0919399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.58 + 0.358i)T \)
good2 \( 1 + (-0.164 - 0.0336i)T + (1.83 + 0.783i)T^{2} \)
3 \( 1 + (1.12 - 0.182i)T + (2.84 - 0.950i)T^{2} \)
5 \( 1 + (0.123 - 0.501i)T + (-4.42 - 2.32i)T^{2} \)
7 \( 1 + (2.88 - 1.37i)T + (4.42 - 5.42i)T^{2} \)
11 \( 1 + (-0.240 + 0.719i)T + (-8.79 - 6.60i)T^{2} \)
17 \( 1 + (-0.268 - 0.565i)T + (-10.7 + 13.1i)T^{2} \)
19 \( 1 + (0.570 + 0.329i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.66 + 6.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.52 - 7.47i)T + (-26.6 - 11.3i)T^{2} \)
31 \( 1 + (0.765 + 1.45i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (-5.47 + 8.65i)T + (-15.8 - 33.4i)T^{2} \)
41 \( 1 + (-1.31 - 8.10i)T + (-38.8 + 12.9i)T^{2} \)
43 \( 1 + (8.14 - 5.14i)T + (18.4 - 38.8i)T^{2} \)
47 \( 1 + (10.3 + 1.25i)T + (45.6 + 11.2i)T^{2} \)
53 \( 1 + (0.232 - 0.336i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (4.85 + 4.66i)T + (2.37 + 58.9i)T^{2} \)
61 \( 1 + (-1.41 + 0.113i)T + (60.2 - 9.78i)T^{2} \)
67 \( 1 + (-1.84 - 4.32i)T + (-46.4 + 48.3i)T^{2} \)
71 \( 1 + (6.31 - 5.15i)T + (14.2 - 69.5i)T^{2} \)
73 \( 1 + (1.09 + 1.23i)T + (-8.79 + 72.4i)T^{2} \)
79 \( 1 + (-0.740 + 6.09i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (11.0 - 4.19i)T + (62.1 - 55.0i)T^{2} \)
89 \( 1 + (8.98 - 5.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.01 + 1.16i)T + (81.9 + 51.8i)T^{2} \)
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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

−12.97476049627862431815476622964, −12.50407970423055231210360344362, −11.22856724154675216090908958666, −10.19394819281650188533661598508, −9.363003627930660599028162426995, −8.324029667015693396150885996391, −6.61791658309277101405614092819, −5.72293238106079440174334058238, −4.68909704964674619506030494101, −3.02291886763891055044279419828, 0.087199298685615539982989988445, 3.23764465560813193928040610102, 4.55635421670868766274473276583, 5.74115788158287987580685865453, 6.96239426832465463630614458334, 8.206744661109874459225795077247, 9.462271229484340141535331368744, 10.08254818528205320618198843880, 11.64690870576113268837867642961, 12.31905592249340136415814530125

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