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Properties

Label 2-350-7.4-c3-0-9
Degree $2$
Conductor $350$
Sign $-0.611 - 0.791i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (3.22 − 5.57i)3-s + (−1.99 + 3.46i)4-s + 12.8·6-s + (3.64 + 18.1i)7-s − 7.99·8-s + (−7.25 − 12.5i)9-s + (−24.1 + 41.9i)11-s + (12.8 + 22.3i)12-s − 93.4·13-s + (−27.8 + 24.4i)14-s + (−8 − 13.8i)16-s + (−10.1 + 17.5i)17-s + (14.5 − 25.1i)18-s + (15.5 + 26.8i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.619 − 1.07i)3-s + (−0.249 + 0.433i)4-s + 0.876·6-s + (0.196 + 0.980i)7-s − 0.353·8-s + (−0.268 − 0.465i)9-s + (−0.663 + 1.14i)11-s + (0.309 + 0.536i)12-s − 1.99·13-s + (−0.530 + 0.467i)14-s + (−0.125 − 0.216i)16-s + (−0.144 + 0.250i)17-s + (0.189 − 0.328i)18-s + (0.187 + 0.324i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.597737821\)
\(L(\frac12)\) \(\approx\) \(1.597737821\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-3.64 - 18.1i)T \)
good3 \( 1 + (-3.22 + 5.57i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (24.1 - 41.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 93.4T + 2.19e3T^{2} \)
17 \( 1 + (10.1 - 17.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-15.5 - 26.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-10.5 - 18.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 69.5T + 2.43e4T^{2} \)
31 \( 1 + (80.5 - 139. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (81.2 + 140. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 365.T + 6.89e4T^{2} \)
43 \( 1 - 254.T + 7.95e4T^{2} \)
47 \( 1 + (-234. - 405. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-293. + 508. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (268. - 465. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-312. - 541. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-61.5 + 106. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 210.T + 3.57e5T^{2} \)
73 \( 1 + (70.8 - 122. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (256. + 444. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 117.T + 5.71e5T^{2} \)
89 \( 1 + (-30.6 - 53.0i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 436.T + 9.12e5T^{2} \)
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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

−12.01920825085651598235133958549, −10.29390234668185367573565661838, −9.274162814643963944736297232172, −8.279612670493037043319604468491, −7.44133791764620917435087602272, −6.95015647342088459234977271541, −5.50499181107439623577586045688, −4.67789066689614435526088014590, −2.76395435952280384247133116966, −1.99049414202521974704235606496, 0.42723349840398216546809046140, 2.52890421931835564161787450460, 3.48682593597346920137420245767, 4.53670911302076058473894018585, 5.27771312483795353502626586598, 6.97128571888190340343645969847, 8.089325435278689014468416371906, 9.179463542613471216574100077518, 10.01962079620495379016588737693, 10.53375815628736321437916422986

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