Properties

Label 2-13e2-13.10-c3-0-15
Degree $2$
Conductor $169$
Sign $0.454 - 0.890i$
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  + (−4.33 + 2.5i)2-s + (3.5 + 6.06i)3-s + (8.50 − 14.7i)4-s − 7i·5-s + (−30.3 − 17.5i)6-s + (−11.2 − 6.5i)7-s + 45.0i·8-s + (−11 + 19.0i)9-s + (17.5 + 30.3i)10-s + (22.5 − 13i)11-s + 119.·12-s + 65·14-s + (42.4 − 24.5i)15-s + (−44.5 − 77.0i)16-s + (38.5 − 66.6i)17-s − 109. i·18-s + ⋯
L(s)  = 1  + (−1.53 + 0.883i)2-s + (0.673 + 1.16i)3-s + (1.06 − 1.84i)4-s − 0.626i·5-s + (−2.06 − 1.19i)6-s + (−0.607 − 0.350i)7-s + 1.98i·8-s + (−0.407 + 0.705i)9-s + (0.553 + 0.958i)10-s + (0.617 − 0.356i)11-s + 2.86·12-s + 1.24·14-s + (0.730 − 0.421i)15-s + (−0.695 − 1.20i)16-s + (0.549 − 0.951i)17-s − 1.44i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.454 - 0.890i$
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: \((2,\ 169,\ (\ :3/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.842310 + 0.515717i\)
\(L(\frac12)\) \(\approx\) \(0.842310 + 0.515717i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.33 - 2.5i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 7iT - 125T^{2} \)
7 \( 1 + (11.2 + 6.5i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-22.5 + 13i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-38.5 + 66.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-109. - 63i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48 + 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-41 - 71.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 196iT - 2.97e4T^{2} \)
37 \( 1 + (-113. + 65.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-290. + 168i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (100.5 - 174. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 105iT - 1.03e5T^{2} \)
53 \( 1 + 432T + 1.48e5T^{2} \)
59 \( 1 + (254. + 147i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-28 + 48.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-413. + 239i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (7.79 + 4.5i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 98iT - 3.89e5T^{2} \)
79 \( 1 - 1.30e3T + 4.93e5T^{2} \)
83 \( 1 + 308iT - 5.71e5T^{2} \)
89 \( 1 + (-1.03e3 + 595i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (60.6 + 35i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.33871407060661981604475077222, −10.86120245812605837674630277374, −9.891013443302997494874154129076, −9.393889044296053743097836492002, −8.666999628385937844236256209911, −7.64056130895943690458729059551, −6.41485914528315961325003745867, −5.00028142789259617936832316768, −3.36545640204411676176985324012, −0.915755285546254250352638949593, 1.10564440381510426738160078380, 2.35721105835190766269507505087, 3.32655968822630159720105257852, 6.37436943234141866849267885162, 7.38669054484479020938787154468, 8.051723985648045370595290696470, 9.227938201009475503870135966098, 9.867022868153539598826201480993, 11.11822618219996700778437450238, 12.01745381500622092154882802454

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