Properties

Label 2-13e2-169.12-c1-0-12
Degree $2$
Conductor $169$
Sign $-0.708 + 0.705i$
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  + (0.651 − 0.450i)2-s + (−1.14 − 0.601i)3-s + (−0.486 + 1.28i)4-s + (−2.65 − 2.99i)5-s + (−1.01 + 0.123i)6-s + (−0.815 − 3.30i)7-s + (0.639 + 2.59i)8-s + (−0.751 − 1.08i)9-s + (−3.07 − 0.758i)10-s + (−0.0864 − 0.0596i)11-s + (1.33 − 1.17i)12-s + (3.60 + 0.129i)13-s + (−2.02 − 1.79i)14-s + (1.23 + 5.03i)15-s + (−0.470 − 0.416i)16-s + (0.528 − 0.130i)17-s + ⋯
L(s)  = 1  + (0.460 − 0.318i)2-s + (−0.662 − 0.347i)3-s + (−0.243 + 0.641i)4-s + (−1.18 − 1.33i)5-s + (−0.415 + 0.0504i)6-s + (−0.308 − 1.25i)7-s + (0.226 + 0.917i)8-s + (−0.250 − 0.362i)9-s + (−0.972 − 0.239i)10-s + (−0.0260 − 0.0179i)11-s + (0.384 − 0.340i)12-s + (0.999 + 0.0359i)13-s + (−0.540 − 0.478i)14-s + (0.320 + 1.29i)15-s + (−0.117 − 0.104i)16-s + (0.128 − 0.0315i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.708 + 0.705i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272536 - 0.659948i\)
\(L(\frac12)\) \(\approx\) \(0.272536 - 0.659948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.60 - 0.129i)T \)
good2 \( 1 + (-0.651 + 0.450i)T + (0.709 - 1.87i)T^{2} \)
3 \( 1 + (1.14 + 0.601i)T + (1.70 + 2.46i)T^{2} \)
5 \( 1 + (2.65 + 2.99i)T + (-0.602 + 4.96i)T^{2} \)
7 \( 1 + (0.815 + 3.30i)T + (-6.19 + 3.25i)T^{2} \)
11 \( 1 + (0.0864 + 0.0596i)T + (3.90 + 10.2i)T^{2} \)
17 \( 1 + (-0.528 + 0.130i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 - 1.09iT - 19T^{2} \)
23 \( 1 + 0.747T + 23T^{2} \)
29 \( 1 + (5.92 + 8.58i)T + (-10.2 + 27.1i)T^{2} \)
31 \( 1 + (-4.48 + 0.544i)T + (30.0 - 7.41i)T^{2} \)
37 \( 1 + (-6.81 + 0.827i)T + (35.9 - 8.85i)T^{2} \)
41 \( 1 + (-5.07 + 9.66i)T + (-23.2 - 33.7i)T^{2} \)
43 \( 1 + (-0.481 + 3.96i)T + (-41.7 - 10.2i)T^{2} \)
47 \( 1 + (-7.74 + 2.93i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (4.21 - 1.03i)T + (46.9 - 24.6i)T^{2} \)
59 \( 1 + (1.30 + 1.47i)T + (-7.11 + 58.5i)T^{2} \)
61 \( 1 + (0.502 + 0.123i)T + (54.0 + 28.3i)T^{2} \)
67 \( 1 + (11.1 - 4.21i)T + (50.1 - 44.4i)T^{2} \)
71 \( 1 + (-0.470 + 0.896i)T + (-40.3 - 58.4i)T^{2} \)
73 \( 1 + (2.57 + 1.77i)T + (25.8 + 68.2i)T^{2} \)
79 \( 1 + (-3.60 - 9.51i)T + (-59.1 + 52.3i)T^{2} \)
83 \( 1 + (-6.17 - 11.7i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + (-5.28 + 5.96i)T + (-11.6 - 96.2i)T^{2} \)
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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

−12.37230942557975618703238244418, −11.69645201396192631379462919642, −10.92967005887042341606527828402, −9.187659364238215345946597629008, −8.155126788397112790415641060976, −7.35344829607224309945178588627, −5.73644850431863730648501310968, −4.26566131472527878460960989412, −3.74843950222783833628859931373, −0.64698901520274580416665965634, 3.05555616828203670846484412220, 4.47160269679466662771100934156, 5.79861918436414293610190318857, 6.44776652492600637768353698974, 7.82717225228647905929340277466, 9.185477618848881849952620219907, 10.54045403901723569845067136277, 11.10750036572007807210779400245, 11.96216293570456607975602238552, 13.20719164013940992217947656900

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