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Properties

Label 2-13e2-13.3-c3-0-6
Degree $2$
Conductor $169$
Sign $0.477 - 0.878i$
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  + (−2.5 − 4.33i)2-s + (3.5 + 6.06i)3-s + (−8.50 + 14.7i)4-s + 7·5-s + (17.5 − 30.3i)6-s + (−6.5 + 11.2i)7-s + 45.0·8-s + (−11 + 19.0i)9-s + (−17.5 − 30.3i)10-s + (−13 − 22.5i)11-s − 119.·12-s + 65·14-s + (24.5 + 42.4i)15-s + (−44.5 − 77.0i)16-s + (−38.5 + 66.6i)17-s + 109.·18-s + ⋯
L(s)  = 1  + (−0.883 − 1.53i)2-s + (0.673 + 1.16i)3-s + (−1.06 + 1.84i)4-s + 0.626·5-s + (1.19 − 2.06i)6-s + (−0.350 + 0.607i)7-s + 1.98·8-s + (−0.407 + 0.705i)9-s + (−0.553 − 0.958i)10-s + (−0.356 − 0.617i)11-s − 2.86·12-s + 1.24·14-s + (0.421 + 0.730i)15-s + (−0.695 − 1.20i)16-s + (−0.549 + 0.951i)17-s + 1.44·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.804048 + 0.478068i\)
\(L(\frac12)\) \(\approx\) \(0.804048 + 0.478068i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 + 4.33i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 7T + 125T^{2} \)
7 \( 1 + (6.5 - 11.2i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (13 + 22.5i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (38.5 - 66.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (63 - 109. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
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 \( 1 + 196T + 2.97e4T^{2} \)
37 \( 1 + (65.5 + 113. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-168 - 290. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-100.5 + 174. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 105T + 1.03e5T^{2} \)
53 \( 1 + 432T + 1.48e5T^{2} \)
59 \( 1 + (147 - 254. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-28 + 48.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-239 - 413. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-4.5 + 7.79i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 98T + 3.89e5T^{2} \)
79 \( 1 - 1.30e3T + 4.93e5T^{2} \)
83 \( 1 - 308T + 5.71e5T^{2} \)
89 \( 1 + (595 + 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-35 + 60.6i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.36032747335644965472207725732, −10.99918459226669365897682717624, −10.41953862586082272335670702926, −9.534210322362023116304212955912, −8.960116505072500222169624362845, −8.112211098299518116727171970876, −5.82733478085236609519846986499, −4.05678550095286923459644667792, −3.12907310242796333162048933211, −1.90016243719302224106282098731, 0.51709899821583641084620976354, 2.26740113364697859585444596896, 4.89114988219898330083302468811, 6.40532387260018605088522162402, 7.02486520333471652254689326860, 7.74579401140566999044386890038, 8.852583368604701797208425053984, 9.577893585456094749499755765687, 10.75639291940734256476841036414, 12.65952483754264354068649741060

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