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Properties

Label 2-13e2-13.5-c2-0-2
Degree $2$
Conductor $169$
Sign $-0.957 - 0.289i$
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  + (−2.34 + 2.34i)2-s − 3·3-s − 7i·4-s + (2.34 − 2.34i)5-s + (7.03 − 7.03i)6-s + (7.03 + 7.03i)7-s + (7.03 + 7.03i)8-s + 11i·10-s + (4.69 + 4.69i)11-s + 21i·12-s − 33·14-s + (−7.03 + 7.03i)15-s − 5·16-s − 3i·17-s + (−14.0 + 14.0i)19-s + (−16.4 − 16.4i)20-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)2-s − 3-s − 1.75i·4-s + (0.469 − 0.469i)5-s + (1.17 − 1.17i)6-s + (1.00 + 1.00i)7-s + (0.879 + 0.879i)8-s + 1.10i·10-s + (0.426 + 0.426i)11-s + 1.75i·12-s − 2.35·14-s + (−0.469 + 0.469i)15-s − 0.312·16-s − 0.176i·17-s + (−0.740 + 0.740i)19-s + (−0.820 − 0.820i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0701373 + 0.473680i\)
\(L(\frac12)\) \(\approx\) \(0.0701373 + 0.473680i\)
\(L(2)\) 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 \( 1 + (2.34 - 2.34i)T - 4iT^{2} \)
3 \( 1 + 3T + 9T^{2} \)
5 \( 1 + (-2.34 + 2.34i)T - 25iT^{2} \)
7 \( 1 + (-7.03 - 7.03i)T + 49iT^{2} \)
11 \( 1 + (-4.69 - 4.69i)T + 121iT^{2} \)
17 \( 1 + 3iT - 289T^{2} \)
19 \( 1 + (14.0 - 14.0i)T - 361iT^{2} \)
23 \( 1 + 12iT - 529T^{2} \)
29 \( 1 + 42T + 841T^{2} \)
31 \( 1 + (28.1 - 28.1i)T - 961iT^{2} \)
37 \( 1 + (-35.1 - 35.1i)T + 1.36e3iT^{2} \)
41 \( 1 + (32.8 - 32.8i)T - 1.68e3iT^{2} \)
43 \( 1 - 49iT - 1.84e3T^{2} \)
47 \( 1 + (2.34 + 2.34i)T + 2.20e3iT^{2} \)
53 \( 1 + 24T + 2.80e3T^{2} \)
59 \( 1 + (-37.5 - 37.5i)T + 3.48e3iT^{2} \)
61 \( 1 - 30T + 3.72e3T^{2} \)
67 \( 1 + (-28.1 + 28.1i)T - 4.48e3iT^{2} \)
71 \( 1 + (-16.4 + 16.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (-28.1 - 28.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 54T + 6.24e3T^{2} \)
83 \( 1 + (-32.8 + 32.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (117. + 117. i)T + 7.92e3iT^{2} \)
97 \( 1 + (14.0 - 14.0i)T - 9.40e3iT^{2} \)
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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

−12.82570242248127954852988067868, −11.79242766999262316119959521244, −10.88003825861519146917911958775, −9.654985932650205836160487403243, −8.805857313211130631884419535493, −7.982852870821903329330192327278, −6.61332342840504900454469836819, −5.71759443496550371006168742915, −4.99055943274169024918634350643, −1.55623987076872935983950675627, 0.50687259975048986983650056356, 2.04687334041393337655454193340, 3.91218329087799270974986489570, 5.60616039871882773883519767843, 7.03095525451888981842280536482, 8.207055330665520210720896398354, 9.341850737456151998212655471552, 10.46013837466436257248400798233, 11.11446930638685808146810898521, 11.40193455686945906529213028459

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