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Properties

Label 2-13e2-13.4-c3-0-20
Degree $2$
Conductor $169$
Sign $0.997 - 0.0771i$
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  + (1.35 + 0.780i)2-s + (−4.34 + 7.52i)3-s + (−2.78 − 4.81i)4-s − 3.56i·5-s + (−11.7 + 6.78i)6-s + (23.5 − 13.5i)7-s − 21.1i·8-s + (−24.2 − 41.9i)9-s + (2.78 − 4.81i)10-s + (13.2 + 7.63i)11-s + 48.3·12-s + 42.4·14-s + (26.7 + 15.4i)15-s + (−5.71 + 9.89i)16-s + (22.2 + 38.5i)17-s − 75.6i·18-s + ⋯
L(s)  = 1  + (0.478 + 0.276i)2-s + (−0.835 + 1.44i)3-s + (−0.347 − 0.602i)4-s − 0.318i·5-s + (−0.799 + 0.461i)6-s + (1.27 − 0.733i)7-s − 0.935i·8-s + (−0.896 − 1.55i)9-s + (0.0879 − 0.152i)10-s + (0.362 + 0.209i)11-s + 1.16·12-s + 0.810·14-s + (0.461 + 0.266i)15-s + (−0.0892 + 0.154i)16-s + (0.317 + 0.550i)17-s − 0.990i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.997 - 0.0771i$
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: \((2,\ 169,\ (\ :3/2),\ 0.997 - 0.0771i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.65110 + 0.0637870i\)
\(L(\frac12)\) \(\approx\) \(1.65110 + 0.0637870i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.35 - 0.780i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (4.34 - 7.52i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 3.56iT - 125T^{2} \)
7 \( 1 + (-23.5 + 13.5i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-13.2 - 7.63i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-22.2 - 38.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-20.7 + 11.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-61.3 + 106. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-109. + 190. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 27.0iT - 2.97e4T^{2} \)
37 \( 1 + (-81.5 - 47.0i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-138. - 80.1i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (75.6 + 131. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 466. iT - 1.03e5T^{2} \)
53 \( 1 + 120.T + 1.48e5T^{2} \)
59 \( 1 + (-380. + 219. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-68.6 - 118. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (443. + 256. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-355. + 205. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 308. iT - 3.89e5T^{2} \)
79 \( 1 + 586.T + 4.93e5T^{2} \)
83 \( 1 - 1.35e3iT - 5.71e5T^{2} \)
89 \( 1 + (-380. - 219. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.30e3 - 755. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.17360640206008598355955119044, −11.07490680362712174757806005173, −10.43711829274642468014545728874, −9.593014968404543686161718385533, −8.411074811634670305532149549461, −6.66260152336343207855536958404, −5.39961148692738702825970799694, −4.67570963010892306993144185832, −4.02418052105701316264836744354, −0.880282749936896532280897402861, 1.38033601424338725115022089696, 2.83557923392185363812666834261, 4.84935027061466530540307304788, 5.71997760604164449008643735860, 7.12388079749635873915802174409, 7.939220283917572145123116398717, 8.942401220435396550097832657036, 11.04610535538273859866192569805, 11.57329174910167473436423229734, 12.26280755739700758618073004005

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