Properties

Label 2-13e2-169.10-c1-0-10
Degree $2$
Conductor $169$
Sign $0.367 + 0.930i$
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  + (0.980 + 0.200i)2-s + (−2.27 + 0.370i)3-s + (−0.918 − 0.391i)4-s + (0.651 − 2.64i)5-s + (−2.30 − 0.0930i)6-s + (4.17 − 1.98i)7-s + (−2.46 − 1.70i)8-s + (2.20 − 0.736i)9-s + (1.16 − 2.46i)10-s + (0.585 − 1.75i)11-s + (2.23 + 0.551i)12-s + (−2.08 + 2.94i)13-s + (4.49 − 1.10i)14-s + (−0.505 + 6.25i)15-s + (−0.697 − 0.726i)16-s + (−2.65 − 5.60i)17-s + ⋯
L(s)  = 1  + (0.693 + 0.141i)2-s + (−1.31 + 0.213i)3-s + (−0.459 − 0.195i)4-s + (0.291 − 1.18i)5-s + (−0.942 − 0.0379i)6-s + (1.57 − 0.749i)7-s + (−0.873 − 0.602i)8-s + (0.734 − 0.245i)9-s + (0.369 − 0.777i)10-s + (0.176 − 0.528i)11-s + (0.645 + 0.159i)12-s + (−0.577 + 0.816i)13-s + (1.20 − 0.296i)14-s + (−0.130 + 1.61i)15-s + (−0.174 − 0.181i)16-s + (−0.645 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.367 + 0.930i$
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: \((2,\ 169,\ (\ :1/2),\ 0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833242 - 0.566906i\)
\(L(\frac12)\) \(\approx\) \(0.833242 - 0.566906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.08 - 2.94i)T \)
good2 \( 1 + (-0.980 - 0.200i)T + (1.83 + 0.783i)T^{2} \)
3 \( 1 + (2.27 - 0.370i)T + (2.84 - 0.950i)T^{2} \)
5 \( 1 + (-0.651 + 2.64i)T + (-4.42 - 2.32i)T^{2} \)
7 \( 1 + (-4.17 + 1.98i)T + (4.42 - 5.42i)T^{2} \)
11 \( 1 + (-0.585 + 1.75i)T + (-8.79 - 6.60i)T^{2} \)
17 \( 1 + (2.65 + 5.60i)T + (-10.7 + 13.1i)T^{2} \)
19 \( 1 + (-0.560 - 0.323i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.07 - 3.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.335 - 1.64i)T + (-26.6 - 11.3i)T^{2} \)
31 \( 1 + (-0.757 - 1.44i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (2.53 - 4.01i)T + (-15.8 - 33.4i)T^{2} \)
41 \( 1 + (-0.803 - 4.94i)T + (-38.8 + 12.9i)T^{2} \)
43 \( 1 + (-7.23 + 4.57i)T + (18.4 - 38.8i)T^{2} \)
47 \( 1 + (-10.4 - 1.26i)T + (45.6 + 11.2i)T^{2} \)
53 \( 1 + (-1.29 + 1.86i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (6.37 + 6.12i)T + (2.37 + 58.9i)T^{2} \)
61 \( 1 + (7.16 - 0.578i)T + (60.2 - 9.78i)T^{2} \)
67 \( 1 + (-3.81 - 8.95i)T + (-46.4 + 48.3i)T^{2} \)
71 \( 1 + (-7.93 + 6.47i)T + (14.2 - 69.5i)T^{2} \)
73 \( 1 + (-0.757 - 0.854i)T + (-8.79 + 72.4i)T^{2} \)
79 \( 1 + (1.56 - 12.8i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (8.62 - 3.27i)T + (62.1 - 55.0i)T^{2} \)
89 \( 1 + (-7.26 + 4.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 + 3.15i)T + (81.9 + 51.8i)T^{2} \)
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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

−12.49425209584842957404345659473, −11.66958639108844787984961266394, −10.96775757864365908418530845596, −9.577387253207613302528417726052, −8.646739727726439198037489316422, −7.06275656387659424177026185572, −5.58928417947527375516460298661, −4.90139743705021609355966300527, −4.41020125291715573191590609292, −0.988917781255826247250863003761, 2.43008409860835093119220172496, 4.41414734016117159781956739406, 5.40976325412232163823700651072, 6.20136659475424529622167058664, 7.60655799791035106844615982153, 8.859621731342472782314112684236, 10.57400522447577772410873372752, 11.06426971424199052287421261325, 12.13172498841473651617399244202, 12.60052101082478168649459090785

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