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Properties

Label 2-13e2-13.11-c2-0-10
Degree $2$
Conductor $169$
Sign $0.693 - 0.720i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.212 − 0.793i)2-s + (2.08 + 3.60i)3-s + (2.87 + 1.66i)4-s + (3.58 + 3.58i)5-s + (3.30 − 0.885i)6-s + (−1.67 − 6.25i)7-s + (4.25 − 4.25i)8-s + (−4.16 + 7.20i)9-s + (3.60 − 2.08i)10-s + (−7.27 − 1.94i)11-s + 13.8i·12-s − 5.32·14-s + (−5.45 + 20.3i)15-s + (4.17 + 7.23i)16-s + (−19.0 − 10.9i)17-s + (4.83 + 4.83i)18-s + ⋯
L(s)  = 1  + (0.106 − 0.396i)2-s + (0.693 + 1.20i)3-s + (0.719 + 0.415i)4-s + (0.716 + 0.716i)5-s + (0.550 − 0.147i)6-s + (−0.239 − 0.893i)7-s + (0.532 − 0.532i)8-s + (−0.462 + 0.801i)9-s + (0.360 − 0.208i)10-s + (−0.661 − 0.177i)11-s + 1.15i·12-s − 0.380·14-s + (−0.363 + 1.35i)15-s + (0.260 + 0.452i)16-s + (−1.11 − 0.646i)17-s + (0.268 + 0.268i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.693 - 0.720i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.693 - 0.720i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.17481 + 0.925623i\)
\(L(\frac12)\) \(\approx\) \(2.17481 + 0.925623i\)
\(L(2)\) 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 \( 1 + (-0.212 + 0.793i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-2.08 - 3.60i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.58 - 3.58i)T + 25iT^{2} \)
7 \( 1 + (1.67 + 6.25i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (7.27 + 1.94i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (19.0 + 10.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.31 - 1.15i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-7.37 + 4.25i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-2.90 - 5.03i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-0.513 - 0.513i)T + 961iT^{2} \)
37 \( 1 + (33.0 + 8.86i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-1.77 + 6.60i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-26.4 - 15.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (37.3 - 37.3i)T - 2.20e3iT^{2} \)
53 \( 1 + 35.8T + 2.80e3T^{2} \)
59 \( 1 + (-21.3 - 79.6i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-40.1 + 69.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-14.3 + 53.3i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (125. - 33.5i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-31.6 + 31.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 18.7T + 6.24e3T^{2} \)
83 \( 1 + (44.6 + 44.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (12.1 + 3.25i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-165. + 44.4i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.83192909117337344069044111580, −11.26741913854993946258211126650, −10.51683559105419073153700701418, −10.04149047679098448704067438389, −8.832421324641564095122857539308, −7.43535764542445814692237272707, −6.42726503279449945308979872658, −4.58150636943364736614417714375, −3.39182676855808148025177079679, −2.45599088617232208746046996207, 1.69313815188650194915360734126, 2.54021476536548748719085569974, 5.15670929871603954637777266331, 6.16148641962633715031319838247, 7.08255796556201153876077129696, 8.242114343949475836197021688472, 9.044335880159985419664829385890, 10.36790571671817524469396209076, 11.68316016956934611540541545135, 12.81263029232965698145683403148

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