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Properties

Label 2-322-7.4-c1-0-10
Degree $2$
Conductor $322$
Sign $0.992 + 0.120i$
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 + (0.186 − 0.323i)3-s + (−0.499 + 0.866i)4-s + (−1.58 − 2.73i)5-s + 0.373·6-s + (2.36 − 1.18i)7-s − 0.999·8-s + (1.43 + 2.47i)9-s + (1.58 − 2.73i)10-s + (1.05 − 1.83i)11-s + (0.186 + 0.323i)12-s + 2.92·13-s + (2.21 + 1.45i)14-s − 1.18·15-s + (−0.5 − 0.866i)16-s + (3.08 − 5.33i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.107 − 0.186i)3-s + (−0.249 + 0.433i)4-s + (−0.707 − 1.22i)5-s + 0.152·6-s + (0.893 − 0.449i)7-s − 0.353·8-s + (0.476 + 0.825i)9-s + (0.500 − 0.866i)10-s + (0.319 − 0.553i)11-s + (0.0539 + 0.0934i)12-s + 0.812·13-s + (0.591 + 0.388i)14-s − 0.305·15-s + (−0.125 − 0.216i)16-s + (0.747 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.992 + 0.120i$
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: \((2,\ 322,\ (\ :1/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60199 - 0.0967384i\)
\(L(\frac12)\) \(\approx\) \(1.60199 - 0.0967384i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 + 1.18i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.186 + 0.323i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.58 + 2.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.05 + 1.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 + (-3.08 + 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0948 + 0.164i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 7.27T + 29T^{2} \)
31 \( 1 + (2.91 - 5.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.69 - 6.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + (-4.45 - 7.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.56 - 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.61 + 9.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.45 + 4.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.573 - 0.993i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + (2.66 - 4.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.19 - 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.05T + 83T^{2} \)
89 \( 1 + (-3.90 - 6.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−11.69746471633842405765561915601, −10.96100424504037229770811094503, −9.456749176340296703232078579751, −8.370100721862759072945431506073, −7.931028453859568697618424822758, −6.96735112761931124652611677192, −5.33889893412971414698061833955, −4.74418173260715118988614443431, −3.61565547089290100557666384613, −1.26669611100894962120646804937, 1.83847671995930325470646366599, 3.49893011166327208093786508804, 4.06397624131547766946806521497, 5.65495167143069387126545729400, 6.76230605127514238164194144041, 7.83804969357247980052614394889, 8.934160031389817274499139030051, 10.07006767020711025663800581219, 10.86600642745871375792894072266, 11.59387181075207028479860191000

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