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Properties

Label 2-13e2-13.3-c1-0-0
Degree $2$
Conductor $169$
Sign $-0.611 - 0.791i$
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  + (1.12 + 1.94i)2-s + (0.277 + 0.480i)3-s + (−1.52 + 2.64i)4-s − 1.44·5-s + (−0.623 + 1.07i)6-s + (−1.02 + 1.77i)7-s − 2.35·8-s + (1.34 − 2.33i)9-s + (−1.62 − 2.81i)10-s + (1.27 + 2.21i)11-s − 1.69·12-s − 4.60·14-s + (−0.400 − 0.694i)15-s + (0.400 + 0.694i)16-s + (2.64 − 4.58i)17-s + 6.04·18-s + ⋯
L(s)  = 1  + (0.794 + 1.37i)2-s + (0.160 + 0.277i)3-s + (−0.762 + 1.32i)4-s − 0.646·5-s + (−0.254 + 0.440i)6-s + (−0.387 + 0.670i)7-s − 0.833·8-s + (0.448 − 0.777i)9-s + (−0.513 − 0.889i)10-s + (0.385 + 0.667i)11-s − 0.488·12-s − 1.23·14-s + (−0.103 − 0.179i)15-s + (0.100 + 0.173i)16-s + (0.642 − 1.11i)17-s + 1.42·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.611 - 0.791i$
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: \((2,\ 169,\ (\ :1/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702135 + 1.43030i\)
\(L(\frac12)\) \(\approx\) \(0.702135 + 1.43030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 1.94i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.277 - 0.480i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + (1.02 - 1.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.27 - 2.21i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.64 + 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.92 + 5.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.945 - 1.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.13 + 1.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.26T + 31T^{2} \)
37 \( 1 + (2.67 + 4.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.637 + 1.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.06 - 5.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.95T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.28 - 7.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.288 + 0.499i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.29 + 3.97i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 7.72T + 83T^{2} \)
89 \( 1 + (3.30 + 5.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.96 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.31293592486676508382710946243, −12.37052849792068554718385194965, −11.56442217790137039574434957359, −9.735891870165948494256853100657, −8.939379714636231942686446275645, −7.51668844787155801207694639329, −6.90642977775259276233257201906, −5.63052577988177386537391055901, −4.51962889787279237510927532994, −3.37132537688323264529287661625, 1.52689883213530529365582765022, 3.37016327596816426240421462671, 4.10040853589825648168981381951, 5.58180037649658538309998755809, 7.25731000500646181093224426441, 8.318256106869341408267950777152, 10.00325808879200294697827716316, 10.57893161583639180765487849826, 11.61041980441695972904060873771, 12.40906186397483256211144051925

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