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Properties

Label 2-350-35.4-c3-0-21
Degree $2$
Conductor $350$
Sign $0.324 + 0.946i$
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 + (6.06 + 17.5i)7-s − 7.99i·8-s + (−13 − 22.5i)9-s + (15 − 25.9i)11-s + (3.46 − 1.99i)12-s − 44i·13-s + (28 + 24.2i)14-s + (−8 − 13.8i)16-s + (20.7 + 12i)17-s + (−45.0 − 26i)18-s + (1 + 1.73i)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 + (0.327 + 0.944i)7-s − 0.353i·8-s + (−0.481 − 0.833i)9-s + (0.411 − 0.712i)11-s + (0.0833 − 0.0481i)12-s − 0.938i·13-s + (0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.296 + 0.171i)17-s + (−0.589 − 0.340i)18-s + (0.0120 + 0.0209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.324 + 0.946i$
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: \((2,\ 350,\ (\ :3/2),\ 0.324 + 0.946i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.852297922\)
\(L(\frac12)\) \(\approx\) \(2.852297922\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.06 - 17.5i)T \)
good3 \( 1 + (-0.866 - 0.5i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-15 + 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 44iT - 2.19e3T^{2} \)
17 \( 1 + (-20.7 - 12i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-158. + 91.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 279T + 2.43e4T^{2} \)
31 \( 1 + (-20 + 34.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (65.8 - 38i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 423T + 6.89e4T^{2} \)
43 \( 1 + 305iT - 7.95e4T^{2} \)
47 \( 1 + (-394. + 228i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (171. + 99i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (231 - 400. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (140.5 + 243. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-432. - 249.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 534T + 3.57e5T^{2} \)
73 \( 1 + (-692. - 400i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (395 + 684. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 597iT - 5.71e5T^{2} \)
89 \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3iT - 9.12e5T^{2} \)
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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

−11.02591019814594392867144718637, −10.11108534990135782961088145669, −8.876611691107975432272936848667, −8.386751876638567735289375028802, −6.74134613568049454577641844261, −5.83780818221719658949007173664, −4.94936668422614855825731103121, −3.46887674337377801432494365093, −2.66344435271861373075420432371, −0.872165827103922320657138288901, 1.53036521612756674777585276989, 3.05545262959840560915943525814, 4.41045044896899023031541625802, 5.09227194055111168856785456192, 6.59514068510083679157138394735, 7.29553031806720890947831812021, 8.203316467166665546439206617089, 9.312923338169031721585676800034, 10.49376039632195264823763740754, 11.33700563179333859090448228799

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