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Properties

Label 2-13e2-13.10-c3-0-23
Degree $2$
Conductor $169$
Sign $0.711 + 0.702i$
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  + (2.59 − 1.5i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − 9i·5-s + (2.59 + 1.5i)6-s + (12.9 + 7.5i)7-s + 21i·8-s + (13 − 22.5i)9-s + (−13.5 − 23.3i)10-s + (41.5 − 24i)11-s + 1.00·12-s + 45·14-s + (7.79 − 4.5i)15-s + (35.5 + 61.4i)16-s + (22.5 − 38.9i)17-s − 78i·18-s + ⋯
L(s)  = 1  + (0.918 − 0.530i)2-s + (0.0962 + 0.166i)3-s + (0.0625 − 0.108i)4-s − 0.804i·5-s + (0.176 + 0.102i)6-s + (0.701 + 0.404i)7-s + 0.928i·8-s + (0.481 − 0.833i)9-s + (−0.426 − 0.739i)10-s + (1.13 − 0.657i)11-s + 0.0240·12-s + 0.859·14-s + (0.134 − 0.0774i)15-s + (0.554 + 0.960i)16-s + (0.321 − 0.555i)17-s − 1.02i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.711 + 0.702i$
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: \((2,\ 169,\ (\ :3/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.83175 - 1.16232i\)
\(L(\frac12)\) \(\approx\) \(2.83175 - 1.16232i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.59 + 1.5i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 9iT - 125T^{2} \)
7 \( 1 + (-12.9 - 7.5i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-41.5 + 24i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-22.5 + 38.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (5.19 + 3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (81 + 140. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-72 - 124. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 264iT - 2.97e4T^{2} \)
37 \( 1 + (262. - 151.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (166. - 96i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-48.5 + 84.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 111iT - 1.03e5T^{2} \)
53 \( 1 + 414T + 1.48e5T^{2} \)
59 \( 1 + (-452. - 261i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (188 - 325. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (31.1 - 18i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (309. + 178.5i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 + 830T + 4.93e5T^{2} \)
83 \( 1 + 438iT - 5.71e5T^{2} \)
89 \( 1 + (-379. + 219i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-737. - 426i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.09536121165718485291301397058, −11.77852302901356327813279647227, −10.39590143436297929330373174892, −8.855986920735463874614860258691, −8.519891328251833280702580905558, −6.66436567562850134121176834660, −5.23623047473452404273853853742, −4.37325363721201500642438261279, −3.22082876825889628708262070850, −1.36040707242220767582168639413, 1.69427528113148729441462086750, 3.76226575345647163945360811346, 4.69886381334394153539946218946, 6.03639941165428366765169790036, 7.06817316862821663289292078068, 7.84528105952122428444716378950, 9.590844670378388749959379948944, 10.47136970503185954407694644136, 11.58151549822521320999375446184, 12.69629204539919882926535957307

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