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Properties

Label 2-847-7.2-c1-0-5
Degree $2$
Conductor $847$
Sign $0.248 - 0.968i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
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  + (0.336 − 0.583i)2-s + (−1.07 − 1.86i)3-s + (0.772 + 1.33i)4-s + (−0.844 + 1.46i)5-s − 1.44·6-s + (−2.51 − 0.817i)7-s + 2.38·8-s + (−0.813 + 1.40i)9-s + (0.569 + 0.985i)10-s + (1.66 − 2.87i)12-s − 3.97·13-s + (−1.32 + 1.19i)14-s + 3.63·15-s + (−0.740 + 1.28i)16-s + (1.28 + 2.22i)17-s + (0.548 + 0.949i)18-s + ⋯
L(s)  = 1  + (0.238 − 0.412i)2-s + (−0.620 − 1.07i)3-s + (0.386 + 0.669i)4-s + (−0.377 + 0.654i)5-s − 0.591·6-s + (−0.951 − 0.309i)7-s + 0.844·8-s + (−0.271 + 0.469i)9-s + (0.179 + 0.311i)10-s + (0.479 − 0.831i)12-s − 1.10·13-s + (−0.354 + 0.318i)14-s + 0.938·15-s + (−0.185 + 0.320i)16-s + (0.311 + 0.539i)17-s + (0.129 + 0.223i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.248 - 0.968i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549023 + 0.426044i\)
\(L(\frac12)\) \(\approx\) \(0.549023 + 0.426044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.51 + 0.817i)T \)
11 \( 1 \)
good2 \( 1 + (-0.336 + 0.583i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.07 + 1.86i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.844 - 1.46i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.97T + 13T^{2} \)
17 \( 1 + (-1.28 - 2.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.167 - 0.289i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.11 - 3.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.11T + 29T^{2} \)
31 \( 1 + (-3.82 - 6.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.07 - 8.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 6.26T + 43T^{2} \)
47 \( 1 + (-1.18 + 2.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.19 - 3.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0174 - 0.0303i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 + 3.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.72 + 4.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.43T + 71T^{2} \)
73 \( 1 + (-0.149 - 0.259i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.11 + 12.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.64T + 83T^{2} \)
89 \( 1 + (6.99 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.30T + 97T^{2} \)
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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

−10.48004239744687592783810242278, −9.888516947516800589063077502846, −8.380318463715353522920235511319, −7.50950067500817680526505050412, −6.86162752581646165564389951975, −6.47668704293092459120198046395, −5.05934442101707898884865654412, −3.64117485016358755673591506607, −2.96009826047952941608701008605, −1.60764598355966462244842981183, 0.32494639247502340281705216652, 2.44340309521313795045792760956, 3.98170471205192584524032627222, 4.87651967204585454555064182758, 5.39502535663111139833009832499, 6.37231536079423138187814093418, 7.20475473176746012788036277272, 8.380834739550719201404187574892, 9.530699240670505254027190503908, 9.998926764884821522056625832303

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