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Properties

Label 2-322-7.4-c1-0-13
Degree $2$
Conductor $322$
Sign $-0.774 + 0.632i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (−0.5 − 0.866i)2-s + (0.563 − 0.975i)3-s + (−0.499 + 0.866i)4-s + (−0.858 − 1.48i)5-s − 1.12·6-s + (0.779 − 2.52i)7-s + 0.999·8-s + (0.865 + 1.49i)9-s + (−0.858 + 1.48i)10-s + (1.82 − 3.16i)11-s + (0.563 + 0.975i)12-s − 3.79·13-s + (−2.57 + 0.589i)14-s − 1.93·15-s + (−0.5 − 0.866i)16-s + (−2.08 + 3.61i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.325 − 0.563i)3-s + (−0.249 + 0.433i)4-s + (−0.383 − 0.664i)5-s − 0.459·6-s + (0.294 − 0.955i)7-s + 0.353·8-s + (0.288 + 0.499i)9-s + (−0.271 + 0.469i)10-s + (0.550 − 0.953i)11-s + (0.162 + 0.281i)12-s − 1.05·13-s + (−0.689 + 0.157i)14-s − 0.499·15-s + (−0.125 − 0.216i)16-s + (−0.506 + 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352959 - 0.990320i\)
\(L(\frac12)\) \(\approx\) \(0.352959 - 0.990320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.779 + 2.52i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.563 + 0.975i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.858 + 1.48i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.82 + 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.79T + 13T^{2} \)
17 \( 1 + (2.08 - 3.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.06 + 7.03i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 2.80T + 29T^{2} \)
31 \( 1 + (4.27 - 7.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.46 + 7.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 8.11T + 43T^{2} \)
47 \( 1 + (-1.47 - 2.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.52 - 2.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.09 - 3.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.65 - 9.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.61 + 11.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + (-2.13 + 3.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.43T + 83T^{2} \)
89 \( 1 + (1.97 + 3.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.121T + 97T^{2} \)
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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

−10.93710108247038157706517981268, −10.73535574711261339982988691720, −9.189745522491717400858744414835, −8.513650633181264891870352955240, −7.60042107934937517874990301275, −6.74981756325710581704374945500, −4.87019046319939083824560049358, −4.00441012442163824843701266131, −2.35886412154525801782841577629, −0.848211961855845004252168154450, 2.30989076152133357370854176806, 3.92733476383879010713317471781, 4.97467168129973175537505836739, 6.34668248360656769033519828969, 7.21210868980448516427448219981, 8.207834751990389450482045437475, 9.353084663115143132244692856671, 9.747464684196289197873414143774, 10.89794731296932183924841764678, 12.01969256736618525247335337242

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