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Properties

Label 2-13e2-13.3-c1-0-4
Degree $2$
Conductor $169$
Sign $0.945 - 0.326i$
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  + (0.866 + 1.5i)2-s + (−1 − 1.73i)3-s + (−0.5 + 0.866i)4-s + 1.73·5-s + (1.73 − 3i)6-s + 1.73·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.59i)10-s + 2·12-s + (−1.73 − 2.99i)15-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s − 1.73·18-s + (−1.73 + 3i)19-s + (−0.866 + 1.50i)20-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.577 − 0.999i)3-s + (−0.250 + 0.433i)4-s + 0.774·5-s + (0.707 − 1.22i)6-s + 0.612·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + 0.577·12-s + (−0.447 − 0.774i)15-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s − 0.408·18-s + (−0.397 + 0.688i)19-s + (−0.193 + 0.335i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48562 + 0.249374i\)
\(L(\frac12)\) \(\approx\) \(1.48562 + 0.249374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.73 - 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + (-4.33 - 7.5i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.59 - 4.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 + 3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.73 + 3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (3.46 + 6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.46 + 6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.03113597301930992250082490013, −12.26874644434234748299709762454, −10.94969145802162550966834362624, −9.866519652568439955142800669401, −8.276433748151106426137030070353, −7.25815811685853860469678321852, −6.15359377268972769751707187780, −5.91962893181555972810241465281, −4.34192001417676835383240037133, −1.80752134955969724882689588861, 2.16477055288425466478804862196, 3.75789500024280062620500480945, 4.82934182953175803873413901625, 5.78011015459741765677317590653, 7.45145844507463007729883656062, 9.293738423359049310958431943775, 9.984192833419730653920066752652, 10.98050381169870328059696612846, 11.44469772120690483190998004542, 12.69852044390338259401356996801

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