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Properties

Label 2-13e2-13.7-c2-0-1
Degree $2$
Conductor $169$
Sign $-0.884 + 0.466i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.52 + 0.944i)2-s + (−1.08 + 1.87i)3-s + (8.07 − 4.66i)4-s + (−0.418 − 0.418i)5-s + (2.04 − 7.62i)6-s + (−1.93 − 0.519i)7-s + (−13.7 + 13.7i)8-s + (2.16 + 3.74i)9-s + (1.87 + 1.08i)10-s + (2.68 + 10.0i)11-s + 20.1i·12-s + 7.32·14-s + (1.23 − 0.331i)15-s + (16.8 − 29.1i)16-s + (−13.8 + 7.98i)17-s + (−11.1 − 11.1i)18-s + ⋯
L(s)  = 1  + (−1.76 + 0.472i)2-s + (−0.360 + 0.624i)3-s + (2.01 − 1.16i)4-s + (−0.0837 − 0.0837i)5-s + (0.340 − 1.27i)6-s + (−0.276 − 0.0741i)7-s + (−1.71 + 1.71i)8-s + (0.240 + 0.416i)9-s + (0.187 + 0.108i)10-s + (0.243 + 0.909i)11-s + 1.68i·12-s + 0.523·14-s + (0.0824 − 0.0221i)15-s + (1.05 − 1.82i)16-s + (−0.813 + 0.469i)17-s + (−0.620 − 0.620i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ -0.884 + 0.466i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0429933 - 0.173808i\)
\(L(\frac12)\) \(\approx\) \(0.0429933 - 0.173808i\)
\(L(2)\) 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 \( 1 + (3.52 - 0.944i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (1.08 - 1.87i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.418 + 0.418i)T + 25iT^{2} \)
7 \( 1 + (1.93 + 0.519i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-2.68 - 10.0i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (13.8 - 7.98i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.15 + 4.31i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (23.8 + 13.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (12.9 - 22.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (19.4 + 19.4i)T + 961iT^{2} \)
37 \( 1 + (-1.54 - 5.77i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-15.2 + 4.08i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (9.97 - 5.75i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (35.3 - 35.3i)T - 2.20e3iT^{2} \)
53 \( 1 + 4.18T + 2.80e3T^{2} \)
59 \( 1 + (41.3 + 11.0i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-33.8 - 58.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (110. - 29.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (18.4 - 68.9i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-31.6 + 31.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 50.7T + 6.24e3T^{2} \)
83 \( 1 + (18.6 + 18.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (33.3 + 124. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (31.9 - 119. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.92734026404353618942010818638, −11.61509186948234538127232769921, −10.64875576463589248215214461290, −10.02033737747570947144598940292, −9.191134263913838868276378148627, −8.108267184892755275195605588976, −7.11052147887703344424778412911, −6.06850335681207027645124436586, −4.45911868179975520468397063610, −1.97683357685559536189432717213, 0.19522840817493442878729903362, 1.70586153202770492188417130458, 3.41689430262510694381191948602, 6.05873395134088100670143431365, 7.01897234179318709943787291857, 7.939748114391092505784258053918, 9.073343775646452045144121430327, 9.764810074301590795687534423353, 11.03564115531802258545807548414, 11.60125754952310532568389535238

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