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Properties

Label 2-322-7.4-c1-0-11
Degree $2$
Conductor $322$
Sign $0.175 + 0.984i$
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 + (1.60 − 2.77i)3-s + (−0.499 + 0.866i)4-s + (1.79 + 3.11i)5-s − 3.20·6-s + (1.82 + 1.91i)7-s + 0.999·8-s + (−3.64 − 6.31i)9-s + (1.79 − 3.11i)10-s + (2.04 − 3.53i)11-s + (1.60 + 2.77i)12-s + 0.932·13-s + (0.741 − 2.53i)14-s + 11.5·15-s + (−0.5 − 0.866i)16-s + (−3.17 + 5.49i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.926 − 1.60i)3-s + (−0.249 + 0.433i)4-s + (0.804 + 1.39i)5-s − 1.30·6-s + (0.691 + 0.722i)7-s + 0.353·8-s + (−1.21 − 2.10i)9-s + (0.569 − 0.985i)10-s + (0.615 − 1.06i)11-s + (0.463 + 0.801i)12-s + 0.258·13-s + (0.198 − 0.678i)14-s + 2.98·15-s + (−0.125 − 0.216i)16-s + (−0.770 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.175 + 0.984i$
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: \((2,\ 322,\ (\ :1/2),\ 0.175 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28611 - 1.07749i\)
\(L(\frac12)\) \(\approx\) \(1.28611 - 1.07749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.91i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.60 + 2.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.79 - 3.11i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.04 + 3.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.932T + 13T^{2} \)
17 \( 1 + (3.17 - 5.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.00 + 3.47i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 5.14T + 29T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.528 + 0.915i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.51T + 41T^{2} \)
43 \( 1 - 0.532T + 43T^{2} \)
47 \( 1 + (-3.63 - 6.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.212 - 0.367i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.384 + 0.666i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.86 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.37 + 4.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 + (-5.69 + 9.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.78 - 3.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + (-5.49 - 9.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−11.29604759201023307812293036680, −10.81294573598748906000726881686, −9.217175672743630705116489072079, −8.648657810924034981504027078770, −7.74436798876818442621881425030, −6.58670621834777267773717901619, −6.04693449547340616207306517053, −3.48700068198902700824780687924, −2.45373070896960965478002241317, −1.69814006284736526650247490555, 1.88620822353276425264777484627, 4.10497606561245842674445661528, 4.67752431440608101438883684371, 5.53403460516559393325592797314, 7.32719162384142502141238924768, 8.458498988582072046065618365980, 9.055479677954687180145243691848, 9.712385821515331426281644974693, 10.36346785449993149005395857495, 11.58646070976175047136245496575

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