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Properties

Label 2-13e2-13.12-c7-0-49
Degree $2$
Conductor $169$
Sign $0.999 + 0.0304i$
Analytic cond. $52.7930$
Root an. cond. $7.26588$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.19i·2-s − 57.0·3-s + 117.·4-s + 431. i·5-s − 182. i·6-s − 225. i·7-s + 785. i·8-s + 1.07e3·9-s − 1.37e3·10-s − 4.16e3i·11-s − 6.72e3·12-s + 719.·14-s − 2.46e4i·15-s + 1.25e4·16-s + 2.58e4·17-s + 3.42e3i·18-s + ⋯
L(s)  = 1  + 0.282i·2-s − 1.22·3-s + 0.920·4-s + 1.54i·5-s − 0.344i·6-s − 0.248i·7-s + 0.542i·8-s + 0.489·9-s − 0.436·10-s − 0.944i·11-s − 1.12·12-s + 0.0700·14-s − 1.88i·15-s + 0.766·16-s + 1.27·17-s + 0.138i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.999 + 0.0304i$
Analytic conductor: \(52.7930\)
Root analytic conductor: \(7.26588\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :7/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.426566853\)
\(L(\frac12)\) \(\approx\) \(1.426566853\)
\(L(\frac{9}{2})\) 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 \( 1 - 3.19iT - 128T^{2} \)
3 \( 1 + 57.0T + 2.18e3T^{2} \)
5 \( 1 - 431. iT - 7.81e4T^{2} \)
7 \( 1 + 225. iT - 8.23e5T^{2} \)
11 \( 1 + 4.16e3iT - 1.94e7T^{2} \)
17 \( 1 - 2.58e4T + 4.10e8T^{2} \)
19 \( 1 + 4.20e4iT - 8.93e8T^{2} \)
23 \( 1 + 7.24e4T + 3.40e9T^{2} \)
29 \( 1 + 1.74e5T + 1.72e10T^{2} \)
31 \( 1 + 2.90e5iT - 2.75e10T^{2} \)
37 \( 1 + 1.32e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.40e5iT - 1.94e11T^{2} \)
43 \( 1 + 1.75e5T + 2.71e11T^{2} \)
47 \( 1 + 4.13e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.28e6T + 1.17e12T^{2} \)
59 \( 1 + 1.35e6iT - 2.48e12T^{2} \)
61 \( 1 - 2.73e6T + 3.14e12T^{2} \)
67 \( 1 - 7.89e5iT - 6.06e12T^{2} \)
71 \( 1 + 2.07e6iT - 9.09e12T^{2} \)
73 \( 1 + 3.16e6iT - 1.10e13T^{2} \)
79 \( 1 - 6.20e6T + 1.92e13T^{2} \)
83 \( 1 - 1.25e5iT - 2.71e13T^{2} \)
89 \( 1 - 1.96e6iT - 4.42e13T^{2} \)
97 \( 1 - 5.34e6iT - 8.07e13T^{2} \)
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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

−11.30457015638466698254214033712, −10.84402428554196987390758000931, −9.909351251970794812988568033451, −7.915379612975953928392229631035, −7.02249667585445518125117615583, −6.17912600943939817939897248388, −5.55727130960444854782397695844, −3.54307890267009814905692846216, −2.37713474308323682708060897668, −0.52363651255751518993174780611, 0.955815508546847644111075177259, 1.85047565975963797455098176828, 3.88079106070405301494446063228, 5.31965227032585903708635270997, 5.81008158071309551458300330669, 7.21776066762593758185422572851, 8.363552238043402513572302325611, 9.797797498455614344160187500529, 10.52009605642865559828452382928, 12.00638075854582445923584832832

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