Properties

Label 2-322-23.12-c1-0-3
Degree $2$
Conductor $322$
Sign $-0.206 - 0.978i$
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  + (−0.841 + 0.540i)2-s + (−0.317 + 2.20i)3-s + (0.415 − 0.909i)4-s + (1.95 − 0.573i)5-s + (−0.925 − 2.02i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−1.88 − 0.554i)9-s + (−1.33 + 1.53i)10-s + (3.96 + 2.55i)11-s + (1.87 + 1.20i)12-s + (0.898 − 1.03i)13-s + (−0.959 − 0.281i)14-s + (0.645 + 4.48i)15-s + (−0.654 − 0.755i)16-s + (−0.290 − 0.637i)17-s + ⋯
L(s)  = 1  + (−0.594 + 0.382i)2-s + (−0.183 + 1.27i)3-s + (0.207 − 0.454i)4-s + (0.872 − 0.256i)5-s + (−0.377 − 0.827i)6-s + (0.247 + 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.629 − 0.184i)9-s + (−0.421 + 0.486i)10-s + (1.19 + 0.768i)11-s + (0.541 + 0.347i)12-s + (0.249 − 0.287i)13-s + (−0.256 − 0.0752i)14-s + (0.166 + 1.15i)15-s + (−0.163 − 0.188i)16-s + (−0.0705 − 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.206 - 0.978i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721114 + 0.889504i\)
\(L(\frac12)\) \(\approx\) \(0.721114 + 0.889504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (4.71 + 0.856i)T \)
good3 \( 1 + (0.317 - 2.20i)T + (-2.87 - 0.845i)T^{2} \)
5 \( 1 + (-1.95 + 0.573i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (-3.96 - 2.55i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-0.898 + 1.03i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (0.290 + 0.637i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.0440 + 0.0965i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.692 + 1.51i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.731 - 5.08i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (8.95 + 2.62i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-6.87 + 2.01i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.0826 + 0.574i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 2.84T + 47T^{2} \)
53 \( 1 + (3.27 + 3.77i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-9.39 + 10.8i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-0.249 - 1.73i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-4.16 + 2.67i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-8.96 + 5.76i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (2.72 - 5.97i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (10.4 - 12.0i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-2.15 - 0.631i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.90 + 13.2i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-15.1 + 4.44i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.61501996468670351211356998496, −10.61949718910198838473755579143, −9.810197106786137532230431096140, −9.335964577323261410101700087383, −8.456739533410668673418951638459, −7.01500159688385077322025347034, −5.88587784698603937568345753673, −5.02049645747821383236449716824, −3.87898722201574075842081127582, −1.84806010897408607257567588641, 1.17859864469636960180561333639, 2.19424205659108371975415240503, 3.89203799730654321833026863603, 5.92492967271778334862318426422, 6.53558388512873852092821823182, 7.52144092514377142278340373735, 8.521100645310652294653988416118, 9.483417904394980849842021458377, 10.46818340336656222347277930262, 11.52076778115850210160476991973

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