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Properties

Label 2-13e2-13.12-c1-0-3
Degree $2$
Conductor $169$
Sign $0.277 - 0.960i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s + 2·3-s − 0.999·4-s − 1.73i·5-s + 3.46i·6-s + 1.73i·8-s + 9-s + 2.99·10-s − 1.99·12-s − 3.46i·15-s − 5·16-s − 3·17-s + 1.73i·18-s − 3.46i·19-s + 1.73i·20-s + ⋯
L(s)  = 1  + 1.22i·2-s + 1.15·3-s − 0.499·4-s − 0.774i·5-s + 1.41i·6-s + 0.612i·8-s + 0.333·9-s + 0.948·10-s − 0.577·12-s − 0.894i·15-s − 1.25·16-s − 0.727·17-s + 0.408i·18-s − 0.794i·19-s + 0.387i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26684 + 0.952866i\)
\(L(\frac12)\) \(\approx\) \(1.26684 + 0.952866i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 8.66iT - 37T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 6.92iT - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−13.44476425929559468346538491569, −12.26818101199900444750312981580, −10.94847564488489828881242774172, −9.332238110505233578204456023620, −8.673371113803740723619786588554, −7.939310022246744940027896702216, −6.86820121374391796497923239343, −5.57603028151523584184682147069, −4.30239335282076746516779963742, −2.44501112123847370636860148022, 2.10572921794485249831794009614, 3.06189891578492420096607173424, 4.11034874790725181817896976209, 6.30914840314477631398284782079, 7.58992859012226347336486584256, 8.735107569485189719239757586320, 9.780212139297250293402784529936, 10.56403233314047195907411460948, 11.52441054296548637032148096270, 12.48263861716559487451818949027

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