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Properties

Label 2-13e2-13.12-c3-0-11
Degree $2$
Conductor $169$
Sign $-0.277 - 0.960i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s + 2·3-s + 5·4-s + 1.73i·5-s + 3.46i·6-s + 13.8i·7-s + 22.5i·8-s − 23·9-s − 2.99·10-s + 13.8i·11-s + 10·12-s − 23.9·14-s + 3.46i·15-s + 1.00·16-s + 117·17-s − 39.8i·18-s + ⋯
L(s)  = 1  + 0.612i·2-s + 0.384·3-s + 0.625·4-s + 0.154i·5-s + 0.235i·6-s + 0.748i·7-s + 0.995i·8-s − 0.851·9-s − 0.0948·10-s + 0.379i·11-s + 0.240·12-s − 0.458·14-s + 0.0596i·15-s + 0.0156·16-s + 1.66·17-s − 0.521i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.26484 + 1.68162i\)
\(L(\frac12)\) \(\approx\) \(1.26484 + 1.68162i\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.73iT - 8T^{2} \)
3 \( 1 - 2T + 27T^{2} \)
5 \( 1 - 1.73iT - 125T^{2} \)
7 \( 1 - 13.8iT - 343T^{2} \)
11 \( 1 - 13.8iT - 1.33e3T^{2} \)
17 \( 1 - 117T + 4.91e3T^{2} \)
19 \( 1 - 114. iT - 6.85e3T^{2} \)
23 \( 1 + 78T + 1.21e4T^{2} \)
29 \( 1 + 141T + 2.43e4T^{2} \)
31 \( 1 + 155. iT - 2.97e4T^{2} \)
37 \( 1 + 143. iT - 5.06e4T^{2} \)
41 \( 1 - 271. iT - 6.89e4T^{2} \)
43 \( 1 - 104T + 7.95e4T^{2} \)
47 \( 1 + 301. iT - 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 - 284. iT - 2.05e5T^{2} \)
61 \( 1 - 145T + 2.26e5T^{2} \)
67 \( 1 + 786. iT - 3.00e5T^{2} \)
71 \( 1 + 1.05e3iT - 3.57e5T^{2} \)
73 \( 1 + 458. iT - 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 - 789. iT - 5.71e5T^{2} \)
89 \( 1 + 976. iT - 7.04e5T^{2} \)
97 \( 1 + 200. iT - 9.12e5T^{2} \)
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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

−12.37886404018165738373650293554, −11.79935189690261994538018618336, −10.62642806607817428357026077402, −9.439198943712432812725712764857, −8.208822071315627447775540158688, −7.58827406840953507209647140960, −6.10536104673807200527334911870, −5.46230330957167140991024366127, −3.35432841855299622498109349817, −2.07957064753648795519160772782, 0.945791144304933191258361455779, 2.69794993348018223144316428941, 3.70528307012963988231683941259, 5.48501557872537541854338693268, 6.85642439129705637202471357487, 7.87215039040839901917117402162, 9.089852905947676091212248003528, 10.22824045953998964852109891144, 11.05134804718197398552149884746, 11.90152420499075338523475973876

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