Properties

Label 2-322-7.4-c1-0-0
Degree $2$
Conductor $322$
Sign $-0.208 - 0.978i$
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 + (−0.873 + 1.51i)3-s + (−0.499 + 0.866i)4-s + (−2.13 − 3.69i)5-s + 1.74·6-s + (1.89 + 1.84i)7-s + 0.999·8-s + (−0.0268 − 0.0464i)9-s + (−2.13 + 3.69i)10-s + (−1.59 + 2.76i)11-s + (−0.873 − 1.51i)12-s − 4.60·13-s + (0.655 − 2.56i)14-s + 7.45·15-s + (−0.5 − 0.866i)16-s + (−1.57 + 2.73i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.504 + 0.873i)3-s + (−0.249 + 0.433i)4-s + (−0.953 − 1.65i)5-s + 0.713·6-s + (0.715 + 0.698i)7-s + 0.353·8-s + (−0.00894 − 0.0154i)9-s + (−0.674 + 1.16i)10-s + (−0.480 + 0.832i)11-s + (−0.252 − 0.436i)12-s − 1.27·13-s + (0.175 − 0.685i)14-s + 1.92·15-s + (−0.125 − 0.216i)16-s + (−0.383 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.208 - 0.978i$
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: \((2,\ 322,\ (\ :1/2),\ -0.208 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263885 + 0.326055i\)
\(L(\frac12)\) \(\approx\) \(0.263885 + 0.326055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 - 1.84i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.873 - 1.51i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.13 + 3.69i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.59 - 2.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
17 \( 1 + (1.57 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.26 - 3.92i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + (1.61 - 2.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.62 - 6.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.11T + 41T^{2} \)
43 \( 1 + 2.29T + 43T^{2} \)
47 \( 1 + (2.84 + 4.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.09 + 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.459 - 0.795i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.74 + 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.37 - 4.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.61T + 71T^{2} \)
73 \( 1 + (-6.68 + 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.71 - 6.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.36T + 83T^{2} \)
89 \( 1 + (-2.61 - 4.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.74T + 97T^{2} \)
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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

−11.90051644284748928401902328407, −11.07467359646037953238541953046, −9.933626026499996611540115389178, −9.289149443332800196415596401642, −8.202009293693185131446737158615, −7.63520134854866240450772186793, −5.23955171852601499216806481748, −4.89816942337439214627295560617, −3.94604990999983482193264610405, −1.83285075804741934406157329409, 0.35390093347535317312634982734, 2.68707538065687514314063399988, 4.27807246801149085385609155524, 5.75798439844904796990316028458, 6.99172984097227342398965082827, 7.29962132318950218293482169057, 7.922858387514333514999440405060, 9.509578706431769710282428227307, 10.77326921462127104492432466542, 11.19901291555712606775214768293

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