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Properties

Label 2-350-35.9-c3-0-13
Degree $2$
Conductor $350$
Sign $0.977 - 0.212i$
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 + (−7.32 + 4.23i)3-s + (1.99 + 3.46i)4-s + 16.9·6-s + (3.87 + 18.1i)7-s − 7.99i·8-s + (22.3 − 38.6i)9-s + (−2.80 − 4.85i)11-s + (−29.3 − 16.9i)12-s − 44.1i·13-s + (11.4 − 35.2i)14-s + (−8 + 13.8i)16-s + (−94.7 + 54.7i)17-s + (−77.2 + 44.6i)18-s + (68.3 − 118. i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.41 + 0.814i)3-s + (0.249 + 0.433i)4-s + 1.15·6-s + (0.209 + 0.977i)7-s − 0.353i·8-s + (0.826 − 1.43i)9-s + (−0.0768 − 0.133i)11-s + (−0.705 − 0.407i)12-s − 0.942i·13-s + (0.217 − 0.672i)14-s + (−0.125 + 0.216i)16-s + (−1.35 + 0.780i)17-s + (−1.01 + 0.584i)18-s + (0.825 − 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.977 - 0.212i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6492921371\)
\(L(\frac12)\) \(\approx\) \(0.6492921371\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.87 - 18.1i)T \)
good3 \( 1 + (7.32 - 4.23i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (2.80 + 4.85i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 44.1iT - 2.19e3T^{2} \)
17 \( 1 + (94.7 - 54.7i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-68.3 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (18.5 + 10.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 99.6T + 2.43e4T^{2} \)
31 \( 1 + (8.56 + 14.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (2.77 + 1.60i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 298.T + 6.89e4T^{2} \)
43 \( 1 + 413. iT - 7.95e4T^{2} \)
47 \( 1 + (-508. - 293. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-520. + 300. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-305. - 529. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (348. - 603. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-401. + 231. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 231.T + 3.57e5T^{2} \)
73 \( 1 + (611. - 352. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-506. + 877. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 476. iT - 5.71e5T^{2} \)
89 \( 1 + (390. - 675. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 908. iT - 9.12e5T^{2} \)
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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

−10.99917864939483688252622747734, −10.43979566736679511012041922450, −9.334810138790535018391029994507, −8.651752910138463948756220378277, −7.23076581876750959570082185877, −6.00645881281247042303567905304, −5.29275369756734514186713326879, −4.13014169726976400504748681936, −2.51610668750296357990471189151, −0.58286315918394951138321958670, 0.68924149834015770134855578942, 1.84018429131494265934265875492, 4.25696201993369636929326042348, 5.40852775555845442873969996124, 6.42766642334121241706695040920, 7.14273785025010097504860438414, 7.80681317538781816333561226414, 9.246749078484683569105366409046, 10.26234617391933581272869210144, 11.16545271905293891435004801722

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