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Properties

Label 2-350-7.4-c3-0-4
Degree $2$
Conductor $350$
Sign $-0.857 - 0.514i$
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 + (−0.5 + 0.866i)3-s + (−1.99 + 3.46i)4-s + 1.99·6-s + (10 + 15.5i)7-s + 7.99·8-s + (13 + 22.5i)9-s + (−17.5 + 30.3i)11-s + (−1.99 − 3.46i)12-s − 66·13-s + (17 − 32.9i)14-s + (−8 − 13.8i)16-s + (29.5 − 51.0i)17-s + (26 − 45.0i)18-s + (−68.5 − 118. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.0962 + 0.166i)3-s + (−0.249 + 0.433i)4-s + 0.136·6-s + (0.539 + 0.841i)7-s + 0.353·8-s + (0.481 + 0.833i)9-s + (−0.479 + 0.830i)11-s + (−0.0481 − 0.0833i)12-s − 1.40·13-s + (0.324 − 0.628i)14-s + (−0.125 − 0.216i)16-s + (0.420 − 0.728i)17-s + (0.340 − 0.589i)18-s + (−0.827 − 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.857 - 0.514i$
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: \((2,\ 350,\ (\ :3/2),\ -0.857 - 0.514i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3842622497\)
\(L(\frac12)\) \(\approx\) \(0.3842622497\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10 - 15.5i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (17.5 - 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 66T + 2.19e3T^{2} \)
17 \( 1 + (-29.5 + 51.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (68.5 + 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 106T + 2.43e4T^{2} \)
31 \( 1 + (37.5 - 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 + 260T + 7.95e4T^{2} \)
47 \( 1 + (85.5 + 148. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (208.5 - 361. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
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 \( 1 + (-219.5 + 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 784T + 3.57e5T^{2} \)
73 \( 1 + (-147.5 + 255. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-247.5 - 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 932T + 5.71e5T^{2} \)
89 \( 1 + (-436.5 - 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 290T + 9.12e5T^{2} \)
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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

−11.44499634760613249600918936028, −10.41816610053329578691005083906, −9.780593995027672180449217878621, −8.782262332110583015717920623212, −7.77771793735576789885569481978, −6.93693587096006716564788222343, −4.96023506311919722308434974509, −4.81390782737162006274418269089, −2.75248657316652977619590646714, −1.92961064648979968451739817522, 0.14923568357506035820431349329, 1.58885509511434552720671401737, 3.60052821760592070351233374361, 4.77837759845928790766213766283, 5.96494565749609450939243639067, 6.92218878783427357446642331818, 7.84131566100644516096246478486, 8.530726450427554341367043069230, 10.00372359237019322727147659437, 10.27313561511752907770263618492

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