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Properties

Label 2-13e2-13.12-c1-0-7
Degree $2$
Conductor $169$
Sign $-0.999 - 0.0304i$
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  − 2.24i·2-s − 0.554·3-s − 3.04·4-s − 1.44i·5-s + 1.24i·6-s − 2.04i·7-s + 2.35i·8-s − 2.69·9-s − 3.24·10-s + 2.55i·11-s + 1.69·12-s − 4.60·14-s + 0.801i·15-s − 0.801·16-s + 5.29·17-s + 6.04i·18-s + ⋯
L(s)  = 1  − 1.58i·2-s − 0.320·3-s − 1.52·4-s − 0.646i·5-s + 0.509i·6-s − 0.774i·7-s + 0.833i·8-s − 0.897·9-s − 1.02·10-s + 0.770i·11-s + 0.488·12-s − 1.23·14-s + 0.207i·15-s − 0.200·16-s + 1.28·17-s + 1.42i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.999 - 0.0304i$
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: \((2,\ 169,\ (\ :1/2),\ -0.999 - 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0133300 + 0.874322i\)
\(L(\frac12)\) \(\approx\) \(0.0133300 + 0.874322i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.24iT - 2T^{2} \)
3 \( 1 + 0.554T + 3T^{2} \)
5 \( 1 + 1.44iT - 5T^{2} \)
7 \( 1 + 2.04iT - 7T^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 + 5.85iT - 19T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 + 4.26iT - 31T^{2} \)
37 \( 1 + 5.35iT - 37T^{2} \)
41 \( 1 - 1.27iT - 41T^{2} \)
43 \( 1 + 6.13T + 43T^{2} \)
47 \( 1 - 2.95iT - 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 - 0.576iT - 67T^{2} \)
71 \( 1 + 4.59iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 7.72iT - 83T^{2} \)
89 \( 1 + 6.61iT - 89T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−12.13758064874736801374663097665, −11.34852619452697244212755496462, −10.48282423448184326122054461446, −9.577238461885718993973492693226, −8.594651049948124948792081288246, −7.09260944398793110736710136572, −5.28550609848476294695048933192, −4.21528380717838058486729973293, −2.77329048613617311691290697780, −0.908308072804752802393651025776, 3.18275912545178646188253779177, 5.26026408374825639984170387691, 5.88986566650745892900077357197, 6.81519974439671424306119761584, 8.120121697823137618007058779479, 8.709148835795785617338107813088, 10.13575814607482411831657565413, 11.37579765486354823250544442888, 12.32288905246754372627812311091, 13.80169752987631717815861157195

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