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Properties

Label 2-13e2-13.10-c1-0-3
Degree $2$
Conductor $169$
Sign $0.542 - 0.839i$
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  + (0.694 − 0.400i)2-s + (1.12 + 1.94i)3-s + (−0.678 + 1.17i)4-s + 0.246i·5-s + (1.56 + 0.900i)6-s + (−2.04 − 1.17i)7-s + 2.69i·8-s + (−1.02 + 1.77i)9-s + (0.0990 + 0.171i)10-s + (3.67 − 2.12i)11-s − 3.04·12-s − 1.89·14-s + (−0.480 + 0.277i)15-s + (−0.277 − 0.480i)16-s + (1.07 − 1.86i)17-s + 1.64i·18-s + ⋯
L(s)  = 1  + (0.491 − 0.283i)2-s + (0.648 + 1.12i)3-s + (−0.339 + 0.587i)4-s + 0.110i·5-s + (0.637 + 0.367i)6-s + (−0.771 − 0.445i)7-s + 0.951i·8-s + (−0.341 + 0.591i)9-s + (0.0313 + 0.0542i)10-s + (1.10 − 0.640i)11-s − 0.880·12-s − 0.505·14-s + (−0.124 + 0.0716i)15-s + (−0.0693 − 0.120i)16-s + (0.261 − 0.453i)17-s + 0.387i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.542 - 0.839i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.542 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37141 + 0.746444i\)
\(L(\frac12)\) \(\approx\) \(1.37141 + 0.746444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.694 + 0.400i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.12 - 1.94i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.246iT - 5T^{2} \)
7 \( 1 + (2.04 + 1.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.07 + 1.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0763 - 0.0440i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.746 - 1.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.31 + 4.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.63iT - 31T^{2} \)
37 \( 1 + (4.92 - 2.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.0 - 5.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.147 - 0.256i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.35iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (5.87 + 3.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.73 - 3.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.65 + 3.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.50 - 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.73iT - 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 - 1.60iT - 83T^{2} \)
89 \( 1 + (-2.49 + 1.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.97 - 4.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.11514815157116585811290239073, −11.96171548979544934121424897080, −10.99958382960110256544920237078, −9.724148061438168450096427018749, −9.133168238029880268610919138975, −8.041310304055127949086736041853, −6.52464580797528817661605755837, −4.83275402187169260054428833096, −3.73943219323885087143754228611, −3.12541254647466532178456321814, 1.59427522493290771386827561409, 3.47098604829777082255236727252, 5.08732310396907272347875386903, 6.51781071014099343246882075686, 6.99794350409076091817376260267, 8.603730554548057399611342813523, 9.346857884727755307350091193230, 10.52502871131118291447753812690, 12.37389889674085707109334020588, 12.58158267318473246259070107724

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