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Properties

Label 2-322-7.4-c1-0-2
Degree $2$
Conductor $322$
Sign $-0.740 - 0.672i$
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 + (−0.271 + 0.469i)3-s + (−0.499 + 0.866i)4-s + (0.298 + 0.516i)5-s − 0.542·6-s + (−2.61 − 0.396i)7-s − 0.999·8-s + (1.35 + 2.34i)9-s + (−0.298 + 0.516i)10-s + (−2.92 + 5.06i)11-s + (−0.271 − 0.469i)12-s + 0.239·13-s + (−0.964 − 2.46i)14-s − 0.323·15-s + (−0.5 − 0.866i)16-s + (1.20 − 2.08i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.156 + 0.271i)3-s + (−0.249 + 0.433i)4-s + (0.133 + 0.231i)5-s − 0.221·6-s + (−0.988 − 0.149i)7-s − 0.353·8-s + (0.451 + 0.781i)9-s + (−0.0943 + 0.163i)10-s + (−0.882 + 1.52i)11-s + (−0.0782 − 0.135i)12-s + 0.0663·13-s + (−0.257 − 0.658i)14-s − 0.0834·15-s + (−0.125 − 0.216i)16-s + (0.291 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.740 - 0.672i$
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: \((2,\ 322,\ (\ :1/2),\ -0.740 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428160 + 1.10810i\)
\(L(\frac12)\) \(\approx\) \(0.428160 + 1.10810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.396i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.271 - 0.469i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.298 - 0.516i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.92 - 5.06i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.239T + 13T^{2} \)
17 \( 1 + (-1.20 + 2.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.54 - 4.41i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 5.19T + 29T^{2} \)
31 \( 1 + (-1.22 + 2.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.33 + 2.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 3.93T + 43T^{2} \)
47 \( 1 + (-3.86 - 6.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.18 - 3.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.08 + 8.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.340 - 0.589i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.56 + 2.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.55T + 71T^{2} \)
73 \( 1 + (-3.30 + 5.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.99 - 5.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.386T + 83T^{2} \)
89 \( 1 + (-6.54 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0666T + 97T^{2} \)
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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

−12.37987739791082299831218078282, −10.88164420491597855010157488347, −9.992805701909746132277251774041, −9.423820491686414426718955339069, −7.71806400932970529113928914302, −7.32173558020164850474600188654, −6.05772265296793036238007741628, −5.05199181143117966638187275508, −4.00665010463842970661847142593, −2.50044826295007814575563977111, 0.78785489604573850298808521694, 2.83570200365886714106661630867, 3.75604305101717251090562701037, 5.38747933908067967035799888049, 6.12994155331944325603510530929, 7.27810154560835916935392739666, 8.745086361773585373317974706908, 9.441934107915564862385389216568, 10.49196105829289016806075682143, 11.31540115595984515708285997484

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