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Properties

Label 2-350-35.4-c1-0-7
Degree $2$
Conductor $350$
Sign $0.669 + 0.742i$
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 + (0.499 − 0.866i)4-s + (2.59 + 0.5i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (1 − 1.73i)11-s + (2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (6.06 + 3.5i)17-s + (−2.59 − 1.5i)18-s − 1.99i·22-s + (−2.59 + 1.5i)23-s + (1.73 − 2i)28-s − 6·29-s + (3.5 − 6.06i)31-s + (−0.866 − 0.499i)32-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.981 + 0.188i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.301 − 0.522i)11-s + (0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (1.47 + 0.848i)17-s + (−0.612 − 0.353i)18-s − 0.426i·22-s + (−0.541 + 0.312i)23-s + (0.327 − 0.377i)28-s − 1.11·29-s + (0.628 − 1.08i)31-s + (−0.153 − 0.0883i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.669 + 0.742i$
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: \((2,\ 350,\ (\ :1/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82188 - 0.810383i\)
\(L(\frac12)\) \(\approx\) \(1.82188 - 0.810383i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-6.06 - 3.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + (6.06 - 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.46 - 2i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (-12.1 - 7i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−11.73580952950270820306793564199, −10.63526951691927470177616735202, −9.637351545592244872028381741092, −8.553023896009665922365098759587, −7.64170337091796259960361270332, −6.15513546850204214880593903738, −5.53576678643747861766465223510, −4.16866772115452463122730823764, −3.13890896507679599036655756324, −1.44984773523033436262610638422, 1.96514011204830873215476330558, 3.51770189566314867206876180782, 4.90789011547799026148635882731, 5.42021458664854185025725445700, 6.89638452408749512590416418684, 7.77570836985041152075369572598, 8.496063414300731290279259296454, 9.872237148049605144564518610235, 10.86660626649499448407734668610, 11.75154790135493148701774041525

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