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Properties

Label 2-13e2-13.9-c1-0-2
Degree $2$
Conductor $169$
Sign $0.562 - 0.826i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.562 - 0.826i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.562 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01888 + 0.538767i\)
\(L(\frac12)\) \(\approx\) \(1.01888 + 0.538767i\)
\(L(\frac{3}{2})\) 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

−12.64993692938241945795896798525, −11.64553167518047477332903085068, −11.41698836570710098794380003264, −10.24979103913571582827894084349, −8.534743960729146402261178762130, −7.978152444558676915709532740344, −6.75766010586965231820362229274, −4.93754204437224256221149326810, −4.02043471144667868502011658411, −2.53560482098662809713636931378, 1.22231433313255160634233390875, 3.81035473216552247170653363370, 4.92524254680606482944363140589, 6.54325330090846404155250528903, 7.20795336139019911466617046865, 8.141622437673912277005388547626, 9.780107012354623370439268608021, 10.85155219470301490305962547160, 11.60292049895524835173811197435, 12.50807141040915472423147121921

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