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Properties

Label 2-322-7.4-c1-0-4
Degree $2$
Conductor $322$
Sign $-0.827 - 0.562i$
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 + (−0.937 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (0.603 + 1.04i)5-s − 1.87·6-s + (2.64 + 0.0264i)7-s − 0.999·8-s + (−0.258 − 0.447i)9-s + (−0.603 + 1.04i)10-s + (−1.40 + 2.43i)11-s + (−0.937 − 1.62i)12-s − 1.15·13-s + (1.29 + 2.30i)14-s − 2.26·15-s + (−0.5 − 0.866i)16-s + (−0.896 + 1.55i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.541 + 0.937i)3-s + (−0.249 + 0.433i)4-s + (0.270 + 0.467i)5-s − 0.765·6-s + (0.999 + 0.0100i)7-s − 0.353·8-s + (−0.0860 − 0.149i)9-s + (−0.191 + 0.330i)10-s + (−0.423 + 0.733i)11-s + (−0.270 − 0.468i)12-s − 0.319·13-s + (0.347 + 0.615i)14-s − 0.584·15-s + (−0.125 − 0.216i)16-s + (−0.217 + 0.376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.827 - 0.562i$
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: \((2,\ 322,\ (\ :1/2),\ -0.827 - 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399424 + 1.29818i\)
\(L(\frac12)\) \(\approx\) \(0.399424 + 1.29818i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.0264i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.937 - 1.62i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.603 - 1.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 + (0.896 - 1.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.20 + 5.54i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 6.14T + 29T^{2} \)
31 \( 1 + (1.52 - 2.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.19 - 7.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.26T + 41T^{2} \)
43 \( 1 - 7.14T + 43T^{2} \)
47 \( 1 + (-2.73 - 4.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.75 + 4.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.61 - 9.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.53 + 4.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.31 - 4.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 + (-5.33 + 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.69 + 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.94T + 83T^{2} \)
89 \( 1 + (3.45 + 5.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.25T + 97T^{2} \)
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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

−11.92234404875755569334765140811, −10.85040870533781771296533314174, −10.39389913426221626477356205188, −9.224392958275366025217069964625, −8.111769061156134148091312163249, −7.08745005227766818731958071981, −5.99911076380507698850324888543, −4.73578502500176130592504879099, −4.49644037192754578681410660952, −2.51725599359964399866706738260, 1.00106522908241647565033960675, 2.26223143565447754276384065967, 4.10063005290357042320431578008, 5.34585765510180392595462489228, 6.06480079970881564887282394692, 7.41501177129990891773657063091, 8.354053118494925464906479763105, 9.454281465780600179466135706275, 10.71569201848239893527457398252, 11.34247041750678353258736271083

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