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Properties

Label 2-13e2-13.10-c3-0-4
Degree $2$
Conductor $169$
Sign $-0.756 - 0.654i$
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  + (−4.17 + 2.41i)2-s + (−2.22 − 3.84i)3-s + (7.64 − 13.2i)4-s + 12.7i·5-s + (18.5 + 10.7i)6-s + (−22.6 − 13.0i)7-s + 35.1i·8-s + (3.62 − 6.28i)9-s + (−30.8 − 53.3i)10-s + (36.6 − 21.1i)11-s − 67.9·12-s + 126.·14-s + (49.1 − 28.3i)15-s + (−23.6 − 41.0i)16-s + (−13.6 + 23.6i)17-s + 35.0i·18-s + ⋯
L(s)  = 1  + (−1.47 + 0.853i)2-s + (−0.427 − 0.740i)3-s + (0.955 − 1.65i)4-s + 1.14i·5-s + (1.26 + 0.729i)6-s + (−1.22 − 0.706i)7-s + 1.55i·8-s + (0.134 − 0.232i)9-s + (−0.974 − 1.68i)10-s + (1.00 − 0.580i)11-s − 1.63·12-s + 2.41·14-s + (0.845 − 0.488i)15-s + (−0.370 − 0.641i)16-s + (−0.194 + 0.337i)17-s + 0.458i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.756 - 0.654i$
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: \((2,\ 169,\ (\ :3/2),\ -0.756 - 0.654i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0916803 + 0.246137i\)
\(L(\frac12)\) \(\approx\) \(0.0916803 + 0.246137i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.17 - 2.41i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (2.22 + 3.84i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 12.7iT - 125T^{2} \)
7 \( 1 + (22.6 + 13.0i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-36.6 + 21.1i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (13.6 - 23.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (11.3 + 6.55i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (14.3 + 24.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-70.8 - 122. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 56.0iT - 2.97e4T^{2} \)
37 \( 1 + (271. - 156. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (305. - 176. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (160. - 277. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 339. iT - 1.03e5T^{2} \)
53 \( 1 - 349.T + 1.48e5T^{2} \)
59 \( 1 + (224. + 129. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (325. - 563. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-775. + 447. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-642. - 370. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 820. iT - 3.89e5T^{2} \)
79 \( 1 + 199.T + 4.93e5T^{2} \)
83 \( 1 + 541. iT - 5.71e5T^{2} \)
89 \( 1 + (-329. + 190. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.23e3 + 715. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.63914184043170918276336981268, −11.35022304010291007674608032808, −10.40187603493186975501074104867, −9.675259649597456284533826601697, −8.531251106297185841079313203515, −7.15152176741046601510002136922, −6.66110863621566200726271909605, −6.22546886456264509420027842410, −3.42883756852499114307389897428, −1.17668283557755262148153682161, 0.23823510264416369386357174067, 1.97374026016650963800964039147, 3.77214513178568627062083157479, 5.26504306792831413432307563688, 6.87662954674959166407474951034, 8.402881811531534166371922441627, 9.265021641307975272005225715636, 9.705610040654778612829544936390, 10.62792594389421565318314109182, 11.95364270462375736344280790547

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