Properties

Label 2-13e2-169.10-c1-0-11
Degree $2$
Conductor $169$
Sign $0.931 - 0.362i$
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  + (2.38 + 0.487i)2-s + (−0.991 + 0.161i)3-s + (3.63 + 1.54i)4-s + (0.415 − 1.68i)5-s + (−2.44 − 0.0986i)6-s + (0.306 − 0.145i)7-s + (3.91 + 2.70i)8-s + (−1.88 + 0.630i)9-s + (1.81 − 3.82i)10-s + (−1.44 + 4.32i)11-s + (−3.85 − 0.949i)12-s + (−0.598 − 3.55i)13-s + (0.804 − 0.198i)14-s + (−0.140 + 1.73i)15-s + (2.56 + 2.66i)16-s + (0.672 + 1.41i)17-s + ⋯
L(s)  = 1  + (1.68 + 0.344i)2-s + (−0.572 + 0.0930i)3-s + (1.81 + 0.774i)4-s + (0.185 − 0.753i)5-s + (−0.999 − 0.0402i)6-s + (0.115 − 0.0550i)7-s + (1.38 + 0.955i)8-s + (−0.629 + 0.210i)9-s + (0.573 − 1.20i)10-s + (−0.435 + 1.30i)11-s + (−1.11 − 0.274i)12-s + (−0.166 − 0.986i)13-s + (0.214 − 0.0529i)14-s + (−0.0362 + 0.448i)15-s + (0.640 + 0.667i)16-s + (0.163 + 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.931 - 0.362i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.931 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21101 + 0.414898i\)
\(L(\frac12)\) \(\approx\) \(2.21101 + 0.414898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.598 + 3.55i)T \)
good2 \( 1 + (-2.38 - 0.487i)T + (1.83 + 0.783i)T^{2} \)
3 \( 1 + (0.991 - 0.161i)T + (2.84 - 0.950i)T^{2} \)
5 \( 1 + (-0.415 + 1.68i)T + (-4.42 - 2.32i)T^{2} \)
7 \( 1 + (-0.306 + 0.145i)T + (4.42 - 5.42i)T^{2} \)
11 \( 1 + (1.44 - 4.32i)T + (-8.79 - 6.60i)T^{2} \)
17 \( 1 + (-0.672 - 1.41i)T + (-10.7 + 13.1i)T^{2} \)
19 \( 1 + (4.58 + 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.471 + 0.816i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.634 + 3.10i)T + (-26.6 - 11.3i)T^{2} \)
31 \( 1 + (-2.06 - 3.92i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (-1.10 + 1.74i)T + (-15.8 - 33.4i)T^{2} \)
41 \( 1 + (0.309 + 1.90i)T + (-38.8 + 12.9i)T^{2} \)
43 \( 1 + (6.96 - 4.40i)T + (18.4 - 38.8i)T^{2} \)
47 \( 1 + (-13.2 - 1.60i)T + (45.6 + 11.2i)T^{2} \)
53 \( 1 + (-0.349 + 0.505i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (-8.15 - 7.83i)T + (2.37 + 58.9i)T^{2} \)
61 \( 1 + (-2.68 + 0.216i)T + (60.2 - 9.78i)T^{2} \)
67 \( 1 + (-4.40 - 10.3i)T + (-46.4 + 48.3i)T^{2} \)
71 \( 1 + (-2.33 + 1.91i)T + (14.2 - 69.5i)T^{2} \)
73 \( 1 + (7.29 + 8.22i)T + (-8.79 + 72.4i)T^{2} \)
79 \( 1 + (-0.217 + 1.79i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (0.165 - 0.0628i)T + (62.1 - 55.0i)T^{2} \)
89 \( 1 + (8.42 - 4.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.1 + 2.95i)T + (81.9 + 51.8i)T^{2} \)
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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

−12.70902411867964900708215094810, −12.39740222114964907535692207024, −11.22449521286546478280719813581, −10.19577030645259701502197330707, −8.499443828396952460942772415305, −7.22317183927328081337483928524, −5.99146167316711883657753264697, −5.12131831134500797326048603547, −4.41994110028155634126965641122, −2.60628589592543009872087943361, 2.50701739307997823795723978942, 3.68386212938536491483990036290, 5.14586033215868716426175883440, 6.08620803119183738573434117224, 6.76529121953505590412069034126, 8.559477336975635959430268969348, 10.36028954465962884315267435810, 11.21858154210057788059759036763, 11.71580773134632248381044834173, 12.73237468545326463068754974952

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