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Properties

Label 2-322-7.2-c1-0-12
Degree $2$
Conductor $322$
Sign $-0.0948 + 0.995i$
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 + (−0.685 − 1.18i)3-s + (−0.499 − 0.866i)4-s + (1.57 − 2.72i)5-s + 1.37·6-s + (−1.66 + 2.05i)7-s + 0.999·8-s + (0.559 − 0.968i)9-s + (1.57 + 2.72i)10-s + (0.126 + 0.219i)11-s + (−0.685 + 1.18i)12-s − 4.85·13-s + (−0.944 − 2.47i)14-s − 4.31·15-s + (−0.5 + 0.866i)16-s + (−3.69 − 6.40i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.395 − 0.685i)3-s + (−0.249 − 0.433i)4-s + (0.702 − 1.21i)5-s + 0.559·6-s + (−0.630 + 0.776i)7-s + 0.353·8-s + (0.186 − 0.322i)9-s + (0.496 + 0.860i)10-s + (0.0381 + 0.0661i)11-s + (−0.197 + 0.342i)12-s − 1.34·13-s + (−0.252 − 0.660i)14-s − 1.11·15-s + (−0.125 + 0.216i)16-s + (−0.896 − 1.55i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0948 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0948 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.0948 + 0.995i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.0948 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531228 - 0.584270i\)
\(L(\frac12)\) \(\approx\) \(0.531228 - 0.584270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 2.05i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.685 + 1.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.57 + 2.72i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.126 - 0.219i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.85T + 13T^{2} \)
17 \( 1 + (3.69 + 6.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.80 + 4.85i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 + (-2.35 - 4.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.73 + 4.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.88T + 41T^{2} \)
43 \( 1 + 5.71T + 43T^{2} \)
47 \( 1 + (-6.06 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.20 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.46 - 9.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.29 - 2.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.78T + 71T^{2} \)
73 \( 1 + (3.20 + 5.55i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (2.68 - 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.98T + 97T^{2} \)
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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

−11.82257538720578794746978285768, −10.01525875735660730621784379449, −9.282319327451872781592761514771, −8.807593914310477125410374808160, −7.26687386066322016357262525586, −6.65714762948344714573077919849, −5.44724776165791727567265154092, −4.83196413933039665107269083180, −2.41497325736596531439504441430, −0.65155021720981193386434625829, 2.16365203793779991820401126993, 3.48127538063966838311230557403, 4.60090738919155219968132901846, 6.08703125098127961835712481139, 7.01173783488788574865665042956, 8.120082778120394312104396189492, 9.808769240921889565307219420484, 10.05188428462958420503592450962, 10.60803493957195171938138141462, 11.53491943618808461856134465465

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