Properties

Label 2-322-23.18-c1-0-5
Degree $2$
Conductor $322$
Sign $0.955 + 0.293i$
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  + (−0.959 − 0.281i)2-s + (−1.90 + 2.19i)3-s + (0.841 + 0.540i)4-s + (0.232 − 1.61i)5-s + (2.44 − 1.56i)6-s + (−0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.771 − 5.36i)9-s + (−0.677 + 1.48i)10-s + (−0.227 + 0.0668i)11-s + (−2.78 + 0.817i)12-s + (1.38 − 3.02i)13-s + (0.142 + 0.989i)14-s + (3.10 + 3.57i)15-s + (0.415 + 0.909i)16-s + (−0.603 + 0.387i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (−1.09 + 1.26i)3-s + (0.420 + 0.270i)4-s + (0.103 − 0.722i)5-s + (0.996 − 0.640i)6-s + (−0.157 − 0.343i)7-s + (−0.231 − 0.267i)8-s + (−0.257 − 1.78i)9-s + (−0.214 + 0.469i)10-s + (−0.0686 + 0.0201i)11-s + (−0.803 + 0.235i)12-s + (0.383 − 0.839i)13-s + (0.0380 + 0.264i)14-s + (0.800 + 0.924i)15-s + (0.103 + 0.227i)16-s + (−0.146 + 0.0940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632546 - 0.0950438i\)
\(L(\frac12)\) \(\approx\) \(0.632546 - 0.0950438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-1.51 + 4.55i)T \)
good3 \( 1 + (1.90 - 2.19i)T + (-0.426 - 2.96i)T^{2} \)
5 \( 1 + (-0.232 + 1.61i)T + (-4.79 - 1.40i)T^{2} \)
11 \( 1 + (0.227 - 0.0668i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.38 + 3.02i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.603 - 0.387i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-5.24 - 3.37i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-5.68 + 3.65i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-3.72 - 4.30i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (1.22 + 8.52i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.825 - 5.74i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-1.05 + 1.21i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.90T + 47T^{2} \)
53 \( 1 + (4.85 + 10.6i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-2.46 + 5.39i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-2.58 - 2.98i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-9.73 - 2.85i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (13.2 + 3.89i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.37 + 1.52i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-3.49 + 7.65i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.0423 - 0.294i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-5.70 + 6.58i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (2.36 - 16.4i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.34427826606784605887791432520, −10.43129208554522531988038074945, −10.00300140494896055281272965861, −9.026632321025987785852491813041, −8.050313254706953388330181068771, −6.56184784089156218548171995638, −5.50334131317512896081273279102, −4.60630817807306085730766775166, −3.34223728208362533378274200985, −0.78751666171325883713642569019, 1.24320060141058019235725151774, 2.76301496397545236293271508278, 5.10428355341282920211466496159, 6.20559858335398505296358871140, 6.83302170867165035123466345884, 7.51777827110286860223943834737, 8.746350165962105274818440280477, 9.888596190766140235843171368591, 11.00850109597074222250429991148, 11.54970201411411438543611205684

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