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Properties

Label 2-13e2-13.3-c1-0-5
Degree $2$
Conductor $169$
Sign $0.309 + 0.950i$
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  + (−0.277 − 0.480i)2-s + (−0.400 − 0.694i)3-s + (0.846 − 1.46i)4-s + 2.80·5-s + (−0.222 + 0.385i)6-s + (−1.34 + 2.33i)7-s − 2.04·8-s + (1.17 − 2.04i)9-s + (−0.777 − 1.34i)10-s + (−0.599 − 1.03i)11-s − 1.35·12-s + 1.49·14-s + (−1.12 − 1.94i)15-s + (−1.12 − 1.94i)16-s + (−0.568 + 0.984i)17-s − 1.30·18-s + ⋯
L(s)  = 1  + (−0.196 − 0.339i)2-s + (−0.231 − 0.400i)3-s + (0.423 − 0.732i)4-s + 1.25·5-s + (−0.0908 + 0.157i)6-s + (−0.508 + 0.881i)7-s − 0.724·8-s + (0.392 − 0.680i)9-s + (−0.245 − 0.425i)10-s + (−0.180 − 0.312i)11-s − 0.391·12-s + 0.399·14-s + (−0.290 − 0.502i)15-s + (−0.280 − 0.486i)16-s + (−0.137 + 0.238i)17-s − 0.308·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.964648 - 0.700458i\)
\(L(\frac12)\) \(\approx\) \(0.964648 - 0.700458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.277 + 0.480i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.400 + 0.694i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.599 + 1.03i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.568 - 0.984i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.969 + 1.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.30 - 3.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.94 - 6.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.89T + 31T^{2} \)
37 \( 1 + (-0.475 - 0.823i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.65 - 2.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.57 - 6.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.69T + 47T^{2} \)
53 \( 1 - 5.87T + 53T^{2} \)
59 \( 1 + (0.00604 - 0.0104i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.01 + 6.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.62 + 8.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.87 - 11.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 0.807T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + (7.36 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.56 + 2.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.56873170274175778258794692011, −11.58773208272517358920206669624, −10.51010566701043853338630885401, −9.537080314376218146678386186460, −9.013817433137607797456300739481, −6.97997442956214812989799427133, −6.09199994723775271278053916526, −5.41710400138799687744647901352, −2.90024203922291450150535493245, −1.51452203325906918332763879181, 2.37179121278572754141533478618, 4.08696175325807710089842869832, 5.59112534631060685168012547097, 6.76109742574085139859311712326, 7.62097003081602881213254990141, 9.017455195732849734338949546261, 10.09671715623067535966500153827, 10.66594265023686621145776693646, 12.05518960405831915121674093057, 13.19907786066282017161892624152

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