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Properties

Label 2-322-7.4-c1-0-7
Degree $2$
Conductor $322$
Sign $0.893 + 0.448i$
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 + (0.313 − 0.543i)3-s + (−0.499 + 0.866i)4-s + (0.475 + 0.823i)5-s − 0.627·6-s + (2.62 − 0.322i)7-s + 0.999·8-s + (1.30 + 2.25i)9-s + (0.475 − 0.823i)10-s + (−1.61 + 2.80i)11-s + (0.313 + 0.543i)12-s + 0.439·13-s + (−1.59 − 2.11i)14-s + 0.596·15-s + (−0.5 − 0.866i)16-s + (2.88 − 4.99i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.181 − 0.313i)3-s + (−0.249 + 0.433i)4-s + (0.212 + 0.368i)5-s − 0.256·6-s + (0.992 − 0.122i)7-s + 0.353·8-s + (0.434 + 0.752i)9-s + (0.150 − 0.260i)10-s + (−0.487 + 0.844i)11-s + (0.0905 + 0.156i)12-s + 0.121·13-s + (−0.425 − 0.564i)14-s + 0.154·15-s + (−0.125 − 0.216i)16-s + (0.699 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.893 + 0.448i$
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: \((2,\ 322,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28727 - 0.304549i\)
\(L(\frac12)\) \(\approx\) \(1.28727 - 0.304549i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.322i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.313 + 0.543i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.475 - 0.823i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.61 - 2.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.439T + 13T^{2} \)
17 \( 1 + (-2.88 + 4.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.89 - 3.28i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 2.06T + 29T^{2} \)
31 \( 1 + (-0.804 + 1.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.65 + 4.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.18T + 41T^{2} \)
43 \( 1 + 2.60T + 43T^{2} \)
47 \( 1 + (-1.23 - 2.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.01 + 1.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.86 - 8.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 12.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.998 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + (-3.63 + 6.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.20 + 7.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + (3.08 + 5.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.280T + 97T^{2} \)
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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

−11.56841242733742905973414596138, −10.43176021285066664954226112305, −10.04624205408968144175731229084, −8.707036669405147235040051019824, −7.68920441387973956597551583441, −7.20156852577102747984631925314, −5.38693306289357175867232577403, −4.39464171557835447870930430947, −2.71634760783572315853018402438, −1.60475036892806967215372319848, 1.35811799748794449148524207937, 3.48508916817792239437860449498, 4.87739520010458514569092763668, 5.72123550416943421683110490748, 6.93231668097060656623165468401, 8.137576637693230022348015349413, 8.693417785158652435269504381914, 9.684897538338892793218947424272, 10.61195905434597645172117402978, 11.54785061057482630567577570957

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