Properties

Label 2-798-133.59-c1-0-12
Degree $2$
Conductor $798$
Sign $0.745 + 0.666i$
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  + (−0.342 − 0.939i)2-s + (0.939 − 0.342i)3-s + (−0.766 + 0.642i)4-s + (−0.893 + 1.06i)5-s + (−0.642 − 0.766i)6-s + (2.63 + 0.266i)7-s + (0.866 + 0.500i)8-s + (0.766 − 0.642i)9-s + (1.30 + 0.475i)10-s + (−1.06 − 1.84i)11-s + (−0.500 + 0.866i)12-s + (−2.48 + 2.08i)13-s + (−0.649 − 2.56i)14-s + (−0.475 + 1.30i)15-s + (0.173 − 0.984i)16-s + (4.52 − 5.39i)17-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (0.542 − 0.197i)3-s + (−0.383 + 0.321i)4-s + (−0.399 + 0.475i)5-s + (−0.262 − 0.312i)6-s + (0.994 + 0.100i)7-s + (0.306 + 0.176i)8-s + (0.255 − 0.214i)9-s + (0.412 + 0.150i)10-s + (−0.321 − 0.557i)11-s + (−0.144 + 0.249i)12-s + (−0.690 + 0.579i)13-s + (−0.173 − 0.685i)14-s + (−0.122 + 0.337i)15-s + (0.0434 − 0.246i)16-s + (1.09 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55726 - 0.595037i\)
\(L(\frac12)\) \(\approx\) \(1.55726 - 0.595037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-2.63 - 0.266i)T \)
19 \( 1 + (-4.04 - 1.63i)T \)
good5 \( 1 + (0.893 - 1.06i)T + (-0.868 - 4.92i)T^{2} \)
11 \( 1 + (1.06 + 1.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.48 - 2.08i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.52 + 5.39i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.558 - 3.16i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-8.74 + 1.54i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 - 0.00653T + 31T^{2} \)
37 \( 1 + (-0.273 + 0.158i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.36 - 4.50i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-10.5 + 3.84i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (8.07 + 9.62i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (2.41 + 2.87i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.645 + 0.541i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.53 - 0.975i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.46 - 6.77i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.14 + 11.3i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.67 - 12.8i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (10.1 + 1.78i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.99 - 1.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.46 + 3.07i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-2.09 + 11.8i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.09032390214673370045067153089, −9.431410864670703348770641518832, −8.461313522489456153656716689945, −7.64807354350197464395685558565, −7.18800163010664690294502764636, −5.53882718513976230949887602706, −4.61195462454580753969548703591, −3.35706169317076682014558511008, −2.58797876961687895470275413582, −1.16881224703155933917902869492, 1.18751909317138349267970967324, 2.80722363067115510220011321289, 4.33273375286060322063697103905, 4.86859594559469766936963974915, 5.91942432028924119155816167033, 7.28574839604931110697241891677, 7.951963268197200896174068198761, 8.353414867281310176990479845188, 9.398812626596988452142316760636, 10.23763788786606041072333092122

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