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Properties

Label 2-13e2-13.11-c2-0-14
Degree $2$
Conductor $169$
Sign $0.490 + 0.871i$
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  + (−0.429 + 1.60i)2-s + (−0.677 − 1.17i)3-s + (1.07 + 0.620i)4-s + (−1.57 − 1.57i)5-s + (2.17 − 0.582i)6-s + (−3.25 − 12.1i)7-s + (−6.15 + 6.15i)8-s + (3.58 − 6.20i)9-s + (3.20 − 1.84i)10-s + (−7.38 − 1.97i)11-s − 1.67i·12-s + 20.9·14-s + (−0.780 + 2.91i)15-s + (−4.74 − 8.22i)16-s + (−12.9 − 7.49i)17-s + (8.41 + 8.41i)18-s + ⋯
L(s)  = 1  + (−0.214 + 0.802i)2-s + (−0.225 − 0.390i)3-s + (0.268 + 0.155i)4-s + (−0.314 − 0.314i)5-s + (0.362 − 0.0970i)6-s + (−0.465 − 1.73i)7-s + (−0.769 + 0.769i)8-s + (0.398 − 0.689i)9-s + (0.320 − 0.184i)10-s + (−0.671 − 0.179i)11-s − 0.139i·12-s + 1.49·14-s + (−0.0520 + 0.194i)15-s + (−0.296 − 0.514i)16-s + (−0.763 − 0.440i)17-s + (0.467 + 0.467i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.490 + 0.871i$
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: \((2,\ 169,\ (\ :1),\ 0.490 + 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.844351 - 0.493455i\)
\(L(\frac12)\) \(\approx\) \(0.844351 - 0.493455i\)
\(L(2)\) 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 \( 1 + (0.429 - 1.60i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (0.677 + 1.17i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.57 + 1.57i)T + 25iT^{2} \)
7 \( 1 + (3.25 + 12.1i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (7.38 + 1.97i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (12.9 + 7.49i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-21.4 + 5.75i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-21.0 + 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.0 - 27.8i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.13 - 1.13i)T + 961iT^{2} \)
37 \( 1 + (1.10 + 0.295i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (6.93 - 25.8i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-2.53 - 1.46i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.5 + 28.5i)T - 2.20e3iT^{2} \)
53 \( 1 + 80.4T + 2.80e3T^{2} \)
59 \( 1 + (4.35 + 16.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (10.8 - 18.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.71 - 13.8i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-4.18 + 1.12i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-26.0 + 26.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 36.3T + 6.24e3T^{2} \)
83 \( 1 + (-56.7 - 56.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-96.2 - 25.7i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-154. + 41.3i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.49052899896780840732472850452, −11.40218602167250379295739658913, −10.41998176007422380990944042526, −9.145851545955451827087438283911, −7.85239452834095536468048288994, −7.08038310896101416137196980258, −6.47313626489025798443723697587, −4.78074351432146988123435687198, −3.24632395194187650220116035113, −0.64236263591886949538325001920, 2.12318917712047719708682656756, 3.22698947342690165039551564011, 5.12608814236645429286118130115, 6.16646124448313934307035731995, 7.57982073003006026323539026793, 9.050133571447039601311357674198, 9.831017922338721237099281245903, 10.84222793049700128838719910817, 11.57003855664544966407648524504, 12.42824521758031186723350492596

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