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Properties

Label 2-350-35.4-c1-0-4
Degree $2$
Conductor $350$
Sign $-0.441 + 0.897i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (−0.866 + 0.5i)2-s + (−2.59 − 1.5i)3-s + (0.499 − 0.866i)4-s + 3·6-s + (2.59 − 0.5i)7-s + 0.999i·8-s + (3 + 5.19i)9-s + (1 − 1.73i)11-s + (−2.59 + 1.50i)12-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s + (−5.19 − 3i)18-s + (−3 − 5.19i)19-s + (−7.5 − 2.59i)21-s + 1.99i·22-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−1.49 − 0.866i)3-s + (0.249 − 0.433i)4-s + 1.22·6-s + (0.981 − 0.188i)7-s + 0.353i·8-s + (1 + 1.73i)9-s + (0.301 − 0.522i)11-s + (−0.749 + 0.433i)12-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s + (−1.22 − 0.707i)18-s + (−0.688 − 1.19i)19-s + (−1.63 − 0.566i)21-s + 0.426i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255871 - 0.410855i\)
\(L(\frac12)\) \(\approx\) \(0.255871 - 0.410855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 + 0.5i)T \)
good3 \( 1 + (2.59 + 1.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.46 + 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.73 - i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.79 + 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−11.22529581688974127745359593341, −10.69086254395004332461974412336, −9.241464219891417370633541770056, −8.180185161112396790357306212199, −7.17785276265171945593343973988, −6.54313216531256782082829026907, −5.49303654544934266054455343763, −4.60640276763200156581279358705, −1.96389021575878599347703612669, −0.50550413380204383980470809741, 1.65367680264202821369297322985, 3.90683987712315382891813101600, 4.81872899830111916020011409958, 5.86910726986214489716337011705, 6.91775225261044768116889079669, 8.212493940436230038301393131032, 9.271438253469550118729966080421, 10.20490110477929618466299674142, 10.86291064267169351384358759973, 11.55043150669961571154952896666

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