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Properties

Label 2-13e2-13.3-c3-0-32
Degree $2$
Conductor $169$
Sign $0.434 - 0.900i$
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  + (−0.863 − 1.49i)2-s + (−3.44 − 5.97i)3-s + (2.50 − 4.34i)4-s − 20.8·5-s + (−5.95 + 10.3i)6-s + (3.78 − 6.55i)7-s − 22.4·8-s + (−10.2 + 17.7i)9-s + (17.9 + 31.1i)10-s + (−2.20 − 3.81i)11-s − 34.5·12-s − 13.0·14-s + (71.8 + 124. i)15-s + (−0.637 − 1.10i)16-s + (36.5 − 63.2i)17-s + 35.5·18-s + ⋯
L(s)  = 1  + (−0.305 − 0.528i)2-s + (−0.663 − 1.14i)3-s + (0.313 − 0.542i)4-s − 1.86·5-s + (−0.405 + 0.702i)6-s + (0.204 − 0.353i)7-s − 0.993·8-s + (−0.380 + 0.659i)9-s + (0.568 + 0.985i)10-s + (−0.0603 − 0.104i)11-s − 0.831·12-s − 0.249·14-s + (1.23 + 2.14i)15-s + (−0.00996 − 0.0172i)16-s + (0.520 − 0.902i)17-s + 0.464·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.229584 + 0.144190i\)
\(L(\frac12)\) \(\approx\) \(0.229584 + 0.144190i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.863 + 1.49i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (3.44 + 5.97i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 20.8T + 125T^{2} \)
7 \( 1 + (-3.78 + 6.55i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (2.20 + 3.81i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-36.5 + 63.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-27.9 + 48.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-16.8 - 29.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (60.7 + 105. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 84.1T + 2.97e4T^{2} \)
37 \( 1 + (-85.8 - 148. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-46.7 - 81.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (220. - 382. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 272.T + 1.03e5T^{2} \)
53 \( 1 + 480.T + 1.48e5T^{2} \)
59 \( 1 + (-175. + 303. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-242. + 419. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (483. + 837. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (201. - 348. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 351.T + 3.89e5T^{2} \)
79 \( 1 + 820.T + 4.93e5T^{2} \)
83 \( 1 + 192.T + 5.71e5T^{2} \)
89 \( 1 + (406. + 704. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-394. + 682. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.49500759701876763412047950454, −11.09285992278888651464820122819, −9.612612856323451674265855368398, −8.070474445774491663991548317992, −7.33113348087291422038566250613, −6.39267391381120830947090912463, −4.84135940777261098536899112619, −3.11588539398628207947916980311, −1.12551374477489632988270786602, −0.17931160111080829701213135013, 3.39750500499994896418464403744, 4.23478137687775221502073494151, 5.57887232507253688042063310773, 7.08747869627737871850252311739, 8.024548652464138171381076500711, 8.803007529745673823168739423874, 10.29417514798187509242338373469, 11.27323644956597990615705422714, 11.89545680582821418090340598699, 12.64322388015282524016070433204

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