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Properties

Label 2-13e2-13.10-c3-0-5
Degree $2$
Conductor $169$
Sign $-0.542 + 0.839i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.19 + 2.42i)2-s + (3.09 + 5.36i)3-s + (7.71 − 13.3i)4-s + 15.2i·5-s + (−25.9 − 14.9i)6-s + (3.73 + 2.15i)7-s + 35.9i·8-s + (−5.69 + 9.87i)9-s + (−36.8 − 63.8i)10-s + (−21.2 + 12.2i)11-s + 95.5·12-s − 20.8·14-s + (−81.7 + 47.2i)15-s + (−25.2 − 43.8i)16-s + (−63.5 + 110. i)17-s − 55.1i·18-s + ⋯
L(s)  = 1  + (−1.48 + 0.855i)2-s + (0.596 + 1.03i)3-s + (0.964 − 1.67i)4-s + 1.36i·5-s + (−1.76 − 1.02i)6-s + (0.201 + 0.116i)7-s + 1.58i·8-s + (−0.211 + 0.365i)9-s + (−1.16 − 2.02i)10-s + (−0.583 + 0.336i)11-s + 2.29·12-s − 0.398·14-s + (−1.40 + 0.812i)15-s + (−0.395 − 0.684i)16-s + (−0.907 + 1.57i)17-s − 0.722i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.542 + 0.839i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.542 + 0.839i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.320573 - 0.588977i\)
\(L(\frac12)\) \(\approx\) \(0.320573 - 0.588977i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.19 - 2.42i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-3.09 - 5.36i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 15.2iT - 125T^{2} \)
7 \( 1 + (-3.73 - 2.15i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (21.2 - 12.2i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (63.5 - 110. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (44.8 + 25.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-43.6 - 75.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (112. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 108. iT - 2.97e4T^{2} \)
37 \( 1 + (-100. + 57.8i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-166. + 95.9i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (61.6 - 106. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 36.7iT - 1.03e5T^{2} \)
53 \( 1 - 119.T + 1.48e5T^{2} \)
59 \( 1 + (-696. - 402. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (339. - 587. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-75.7 + 43.7i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (849. + 490. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 263. iT - 3.89e5T^{2} \)
79 \( 1 - 321.T + 4.93e5T^{2} \)
83 \( 1 - 1.04e3iT - 5.71e5T^{2} \)
89 \( 1 + (298. - 172. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (417. + 241. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.23392117913447492359402143556, −11.20474084376853122806316872067, −10.52416138465100170265672541143, −9.875687730050443514762546693425, −8.926857413986419347296731077745, −7.987551397868678271869426972553, −6.97878895598806606552617824149, −5.94815648380716867600859478765, −4.00954500233048247485572289316, −2.32582538006756667551818330907, 0.46499110487462341800026382611, 1.57502076163710117051522938673, 2.75737012293507170165445873017, 4.90266709115361732618985257490, 7.00140474759833085777938959553, 7.954568286702378422569213570998, 8.677802136634065505063064732409, 9.271646706004747293052084935028, 10.61294722230804499455207326028, 11.59302174975008077418271824626

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