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Properties

Label 2-616-7.4-c1-0-10
Degree $2$
Conductor $616$
Sign $0.900 + 0.435i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
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  + (0.207 − 0.358i)3-s + (−0.707 − 1.22i)5-s + (2.62 − 0.358i)7-s + (1.41 + 2.44i)9-s + (−0.5 + 0.866i)11-s + 1.82·13-s − 0.585·15-s + (1 − 1.73i)17-s + (1.29 + 2.23i)19-s + (0.414 − 1.01i)21-s + (−2.70 − 4.68i)23-s + (1.50 − 2.59i)25-s + 2.41·27-s + 29-s + (4.82 − 8.36i)31-s + ⋯
L(s)  = 1  + (0.119 − 0.207i)3-s + (−0.316 − 0.547i)5-s + (0.990 − 0.135i)7-s + (0.471 + 0.816i)9-s + (−0.150 + 0.261i)11-s + 0.507·13-s − 0.151·15-s + (0.242 − 0.420i)17-s + (0.296 + 0.513i)19-s + (0.0903 − 0.221i)21-s + (−0.564 − 0.977i)23-s + (0.300 − 0.519i)25-s + 0.464·27-s + 0.185·29-s + (0.867 − 1.50i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66247 - 0.381350i\)
\(L(\frac12)\) \(\approx\) \(1.66247 - 0.381350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.707 + 1.22i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.82T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.29 - 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.70 + 4.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-4.82 + 8.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.53 - 7.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (2.41 + 4.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.29 - 2.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.20 - 3.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.08 - 7.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.03 - 6.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.75T + 71T^{2} \)
73 \( 1 + (-1.70 + 2.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.03 + 6.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (-4.58 - 7.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−10.51838021447857389900070743636, −9.829841004242472955804737227505, −8.427966340702323392312351432029, −8.113696559445073871615338317716, −7.23268746270647485100722028442, −5.97590244578886457962677390756, −4.78094598770218405033849484377, −4.26375492630593872601243152019, −2.50922963941114612604574900415, −1.20843327737567235018179603868, 1.38665609045316762860391163589, 3.06195983504856918908553014603, 3.99514800990315884886387213557, 5.11110098458588594968917133374, 6.22134086282649029077306141139, 7.20715712399365166480265638220, 8.057220221667309618430774149333, 8.924847772513166191109252040866, 9.838785910006303301713622616263, 10.82390130826101612090341864707

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