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Properties

Label 2-798-7.4-c1-0-6
Degree $2$
Conductor $798$
Sign $0.827 - 0.561i$
Analytic cond. $6.37206$
Root an. cond. $2.52429$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (2.20 + 3.82i)5-s − 0.999·6-s + (1.62 + 2.09i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (2.20 − 3.82i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s − 2.82·13-s + (0.999 − 2.44i)14-s + 4.41·15-s + (−0.5 − 0.866i)16-s + (−2.12 + 3.67i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.987 + 1.70i)5-s − 0.408·6-s + (0.612 + 0.790i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.697 − 1.20i)10-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s − 0.784·13-s + (0.267 − 0.654i)14-s + 1.13·15-s + (−0.125 − 0.216i)16-s + (−0.514 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53214 + 0.470781i\)
\(L(\frac12)\) \(\approx\) \(1.53214 + 0.470781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.62 - 2.09i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-2.20 - 3.82i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + (2.12 - 3.67i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.292 + 0.507i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.82T + 29T^{2} \)
31 \( 1 + (0.328 - 0.568i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.12 - 7.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 + (4.41 + 7.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.91 + 3.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.414 + 0.717i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + (-4.58 + 7.94i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.91 - 5.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.41T + 97T^{2} \)
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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

−10.29121849949318155703928642187, −9.721780270303666700972248371827, −8.679813850381861875854376969483, −7.922352247487783809120963116568, −6.78004144603795916847284650170, −6.27734324818097041767077128382, −5.03098342559863342097605207453, −3.38285843419802107568911980762, −2.48752745332071949595791546200, −1.85416925279357554130139184771, 0.894268635537305127880504935690, 2.19838035359858025859853378702, 4.31731730877847880282540004773, 4.80149785452545287022380112876, 5.55812128442306022070008673540, 6.77167011765782445392694714894, 7.87624285662779621647626394930, 8.526381875437665324705525069757, 9.409915357832134564743516490228, 9.774938809319603898872846878857

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