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Properties

Label 2-13e2-13.5-c2-0-19
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

\(L(\frac{3}{2})\) \(\approx\) \(0.154221 + 1.04155i\)
\(L(\frac12)\) \(\approx\) \(0.154221 + 1.04155i\)
\(L(2)\) 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

−11.92279987710218643222001123195, −11.06994415721236317355348851382, −10.70067068988378687300326735610, −9.556270926480331721400202404721, −7.44521390189786916926397691729, −6.25983510352376037865786824071, −5.21341032749765676133468647104, −3.92066980835038460154751971117, −2.92801790437626255193970967463, −0.48741489902242836634800069835, 3.29982501017726773349076263126, 4.79416373267360741953969097080, 5.64795850130428998962826749704, 6.32901422867944884315413789571, 7.51891451909267780576667476718, 8.708148978305236828187232442431, 10.13345105194259595612582225049, 11.75621861672356443608747779517, 12.29089755777210887032231574552, 12.95919469391668371063399769581

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