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Properties

Label 2-13e2-13.7-c2-0-0
Degree $2$
Conductor $169$
Sign $-0.999 - 0.0386i$
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  + (1.86 − 0.5i)2-s + (−1.36 + 2.36i)3-s + (−0.232 + 0.133i)4-s + (−4.36 − 4.36i)5-s + (−1.36 + 5.09i)6-s + (−8.46 − 2.26i)7-s + (−5.83 + 5.83i)8-s + (0.767 + 1.33i)9-s + (−10.3 − 5.96i)10-s + (−1.66 − 6.19i)11-s − 0.732i·12-s − 16.9·14-s + (16.2 − 4.36i)15-s + (−7.42 + 12.8i)16-s + (−9.99 + 5.76i)17-s + (2.09 + 2.09i)18-s + ⋯
L(s)  = 1  + (0.933 − 0.250i)2-s + (−0.455 + 0.788i)3-s + (−0.0580 + 0.0334i)4-s + (−0.873 − 0.873i)5-s + (−0.227 + 0.849i)6-s + (−1.20 − 0.323i)7-s + (−0.728 + 0.728i)8-s + (0.0853 + 0.147i)9-s + (−1.03 − 0.596i)10-s + (−0.150 − 0.563i)11-s − 0.0610i·12-s − 1.20·14-s + (1.08 − 0.291i)15-s + (−0.464 + 0.804i)16-s + (−0.587 + 0.339i)17-s + (0.116 + 0.116i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0386i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.999 - 0.0386i$
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: \((2,\ 169,\ (\ :1),\ -0.999 - 0.0386i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00284372 + 0.147272i\)
\(L(\frac12)\) \(\approx\) \(0.00284372 + 0.147272i\)
\(L(2)\) 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 \( 1 + (-1.86 + 0.5i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (1.36 - 2.36i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.36 + 4.36i)T + 25iT^{2} \)
7 \( 1 + (8.46 + 2.26i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (1.66 + 6.19i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (9.99 - 5.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.901 - 3.36i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-8.49 - 4.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.69 + 9.86i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.92 - 1.92i)T + 961iT^{2} \)
37 \( 1 + (-11.2 - 42.1i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (18.9 - 5.08i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (45 - 25.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-0.320 + 0.320i)T - 2.20e3iT^{2} \)
53 \( 1 - 78.7T + 2.80e3T^{2} \)
59 \( 1 + (40.9 + 10.9i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (49.1 + 85.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (74.5 - 19.9i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-8.31 + 31.0i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-48.2 + 48.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 82.7T + 6.24e3T^{2} \)
83 \( 1 + (69.5 + 69.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (-8.52 - 31.8i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (20.0 - 74.8i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−13.11169581328945215783945484128, −12.13867895283604195582766046739, −11.31279479874529629660681814009, −10.18689702987457682406303694853, −9.056985141389933427793298634278, −8.030006027346294769527341342693, −6.30146773256022371226131316793, −5.04158435051291363416355614481, −4.23582386098910930144048075870, −3.29803238534585579377590398981, 0.06837857285609762124790845011, 2.98774279935911102777837256144, 4.15112555783857483803076659912, 5.72005416268390527324805610504, 6.77923525477261594378088809584, 7.16635778029604075185826848571, 9.025665508939271000383754500784, 10.13887414639356950599861903498, 11.49061354580289696454606204226, 12.35764482054200563066702355643

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