Properties

Label 2-13e2-169.12-c1-0-5
Degree $2$
Conductor $169$
Sign $0.658 - 0.752i$
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  + (0.197 − 0.136i)2-s + (1.40 + 0.735i)3-s + (−0.688 + 1.81i)4-s + (−0.369 − 0.416i)5-s + (0.377 − 0.0457i)6-s + (0.397 + 1.61i)7-s + (0.226 + 0.918i)8-s + (−0.280 − 0.406i)9-s + (−0.129 − 0.0319i)10-s + (3.54 + 2.44i)11-s + (−2.30 + 2.03i)12-s + (0.00632 − 3.60i)13-s + (0.298 + 0.264i)14-s + (−0.210 − 0.855i)15-s + (−2.73 − 2.42i)16-s + (−3.12 + 0.770i)17-s + ⋯
L(s)  = 1  + (0.139 − 0.0964i)2-s + (0.809 + 0.424i)3-s + (−0.344 + 0.908i)4-s + (−0.165 − 0.186i)5-s + (0.153 − 0.0186i)6-s + (0.150 + 0.608i)7-s + (0.0800 + 0.324i)8-s + (−0.0936 − 0.135i)9-s + (−0.0410 − 0.0101i)10-s + (1.06 + 0.738i)11-s + (−0.664 + 0.588i)12-s + (0.00175 − 0.999i)13-s + (0.0796 + 0.0705i)14-s + (−0.0544 − 0.220i)15-s + (−0.684 − 0.606i)16-s + (−0.758 + 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30069 + 0.589928i\)
\(L(\frac12)\) \(\approx\) \(1.30069 + 0.589928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.00632 + 3.60i)T \)
good2 \( 1 + (-0.197 + 0.136i)T + (0.709 - 1.87i)T^{2} \)
3 \( 1 + (-1.40 - 0.735i)T + (1.70 + 2.46i)T^{2} \)
5 \( 1 + (0.369 + 0.416i)T + (-0.602 + 4.96i)T^{2} \)
7 \( 1 + (-0.397 - 1.61i)T + (-6.19 + 3.25i)T^{2} \)
11 \( 1 + (-3.54 - 2.44i)T + (3.90 + 10.2i)T^{2} \)
17 \( 1 + (3.12 - 0.770i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 + 1.72iT - 19T^{2} \)
23 \( 1 - 7.98T + 23T^{2} \)
29 \( 1 + (0.395 + 0.573i)T + (-10.2 + 27.1i)T^{2} \)
31 \( 1 + (4.11 - 0.499i)T + (30.0 - 7.41i)T^{2} \)
37 \( 1 + (1.44 - 0.175i)T + (35.9 - 8.85i)T^{2} \)
41 \( 1 + (0.456 - 0.868i)T + (-23.2 - 33.7i)T^{2} \)
43 \( 1 + (-0.325 + 2.67i)T + (-41.7 - 10.2i)T^{2} \)
47 \( 1 + (11.5 - 4.39i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (1.84 - 0.454i)T + (46.9 - 24.6i)T^{2} \)
59 \( 1 + (-3.10 - 3.50i)T + (-7.11 + 58.5i)T^{2} \)
61 \( 1 + (9.59 + 2.36i)T + (54.0 + 28.3i)T^{2} \)
67 \( 1 + (4.31 - 1.63i)T + (50.1 - 44.4i)T^{2} \)
71 \( 1 + (-3.75 + 7.14i)T + (-40.3 - 58.4i)T^{2} \)
73 \( 1 + (-10.7 - 7.45i)T + (25.8 + 68.2i)T^{2} \)
79 \( 1 + (-4.76 - 12.5i)T + (-59.1 + 52.3i)T^{2} \)
83 \( 1 + (-4.25 - 8.11i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 + (-3.54 + 4.00i)T + (-11.6 - 96.2i)T^{2} \)
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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

−12.80863538528174141070311408910, −12.15263926833023451728060366892, −11.08110324771287642520786480599, −9.484424625481011853208137960309, −8.874144056028570955495266893604, −8.091933642273486531191528260638, −6.75318125104102340506109814898, −4.93740970465429818203112366509, −3.81633774504160160194113902132, −2.67640115155762614251790696895, 1.58448326007571607951666408416, 3.55648077447665411005779160768, 4.91053587615605023384374953974, 6.44210280687976747746842354563, 7.31777765640432099068887613308, 8.802304250788184315133429292552, 9.291187514243091114092932273210, 10.81370187791011812341927935483, 11.42116392266414121514779995606, 13.12205090725684945060428723896

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