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Properties

Label 2-13e2-13.4-c3-0-9
Degree $2$
Conductor $169$
Sign $-0.743 - 0.669i$
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  + (3.32 + 1.91i)2-s + (0.139 − 0.241i)3-s + (3.35 + 5.80i)4-s + 11.3i·5-s + (0.927 − 0.535i)6-s + (−26.9 + 15.5i)7-s − 4.97i·8-s + (13.4 + 23.3i)9-s + (−21.8 + 37.7i)10-s + (−18.1 − 10.4i)11-s + 1.87·12-s − 119.·14-s + (2.74 + 1.58i)15-s + (36.3 − 62.9i)16-s + (57.1 + 99.0i)17-s + 103. i·18-s + ⋯
L(s)  = 1  + (1.17 + 0.677i)2-s + (0.0268 − 0.0465i)3-s + (0.418 + 0.725i)4-s + 1.01i·5-s + (0.0630 − 0.0364i)6-s + (−1.45 + 0.838i)7-s − 0.219i·8-s + (0.498 + 0.863i)9-s + (−0.689 + 1.19i)10-s + (−0.497 − 0.287i)11-s + 0.0450·12-s − 2.27·14-s + (0.0473 + 0.0273i)15-s + (0.567 − 0.983i)16-s + (0.815 + 1.41i)17-s + 1.35i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.743 - 0.669i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.743 - 0.669i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.870990 + 2.26939i\)
\(L(\frac12)\) \(\approx\) \(0.870990 + 2.26939i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.32 - 1.91i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-0.139 + 0.241i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 11.3iT - 125T^{2} \)
7 \( 1 + (26.9 - 15.5i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (18.1 + 10.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-57.1 - 99.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (39.1 - 22.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-36.9 + 64.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-13.6 + 23.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 179. iT - 2.97e4T^{2} \)
37 \( 1 + (-307. - 177. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (70.5 + 40.7i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (128. + 221. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 463. iT - 1.03e5T^{2} \)
53 \( 1 - 76.6T + 1.48e5T^{2} \)
59 \( 1 + (47.1 - 27.2i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-247. - 428. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-530. - 305. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-14.1 + 8.15i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 321. iT - 3.89e5T^{2} \)
79 \( 1 - 385.T + 4.93e5T^{2} \)
83 \( 1 - 663. iT - 5.71e5T^{2} \)
89 \( 1 + (-472. - 272. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (596. - 344. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.92422120086853067698690490716, −12.22517293107013342082178088066, −10.56543538604403975327811597372, −9.978471786456924360992646126382, −8.316114156733208312153523810077, −6.96855045065851247736020628988, −6.28290698104009240027089638156, −5.33686431646176209655083357237, −3.73541087778878483176074775430, −2.70456521202334592467998040739, 0.77842135707770566355048366756, 2.94349353383378409203331955385, 3.99760285568021695625102765016, 4.99145751551671900045701175921, 6.30435130379882108328425277519, 7.59975042883381721109905513547, 9.289857587717566461981226117569, 9.906849937853548409420027271152, 11.26018078180735310850503276935, 12.37506456027452001151482618373

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