Properties

Label 2-350-35.4-c3-0-6
Degree $2$
Conductor $350$
Sign $0.0764 - 0.997i$
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  + (1.73 − i)2-s + (0.866 + 0.5i)3-s + (1.99 − 3.46i)4-s + 1.99·6-s + (−15.5 + 10i)7-s − 7.99i·8-s + (−13 − 22.5i)9-s + (−17.5 + 30.3i)11-s + (3.46 − 1.99i)12-s + 66i·13-s + (−17 + 32.9i)14-s + (−8 − 13.8i)16-s + (51.0 + 29.5i)17-s + (−45.0 − 26i)18-s + (68.5 + 118. i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.166 + 0.0962i)3-s + (0.249 − 0.433i)4-s + 0.136·6-s + (−0.841 + 0.539i)7-s − 0.353i·8-s + (−0.481 − 0.833i)9-s + (−0.479 + 0.830i)11-s + (0.0833 − 0.0481i)12-s + 1.40i·13-s + (−0.324 + 0.628i)14-s + (−0.125 − 0.216i)16-s + (0.728 + 0.420i)17-s + (−0.589 − 0.340i)18-s + (0.827 + 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0764 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0764 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.0764 - 0.997i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.0764 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.622008663\)
\(L(\frac12)\) \(\approx\) \(1.622008663\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
5 \( 1 \)
7 \( 1 + (15.5 - 10i)T \)
good3 \( 1 + (-0.866 - 0.5i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (17.5 - 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 66iT - 2.19e3T^{2} \)
17 \( 1 + (-51.0 - 29.5i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-68.5 - 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.06 - 3.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (37.5 - 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (9.52 - 5.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 - 260iT - 7.95e4T^{2} \)
47 \( 1 + (-148. + 85.5i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-361. - 208.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (8.5 - 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (25.5 + 44.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-380. - 219.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 784T + 3.57e5T^{2} \)
73 \( 1 + (255. + 147.5i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (247.5 + 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 932iT - 5.71e5T^{2} \)
89 \( 1 + (436.5 + 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 290iT - 9.12e5T^{2} \)
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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

−11.68958379348184726509691263928, −10.18043517148651945636199502754, −9.647965037873032369914177231608, −8.696235033586939701927111652045, −7.28648079424331449098567036291, −6.26662014020823614078464123561, −5.41853196441965089919258327920, −3.99259331391546056752027926353, −3.11981571549896354618729587627, −1.73007870995967901252662077101, 0.43366442329176320547944910563, 2.76202498214661145144693748068, 3.44933552091588508124808041294, 5.12853803090570466999503606837, 5.72661511651931069529236329537, 7.09258563073147450446089328323, 7.79053627591157307311196968650, 8.783158204542505854684111764667, 10.07552002784464697587267633603, 10.86928845001454176746920797981

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