Properties

Label 2-350-175.4-c1-0-8
Degree $2$
Conductor $350$
Sign $0.992 - 0.121i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.615 − 1.38i)3-s + (−0.913 + 0.406i)4-s + (−1.47 + 1.68i)5-s + (1.22 − 0.889i)6-s + (−0.0303 − 2.64i)7-s + (−0.587 − 0.809i)8-s + (0.473 − 0.525i)9-s + (−1.95 − 1.09i)10-s + (4.14 + 4.60i)11-s + (1.12 + 1.01i)12-s + (6.53 − 2.12i)13-s + (2.58 − 0.579i)14-s + (3.23 + 1.00i)15-s + (0.669 − 0.743i)16-s + (5.62 − 0.591i)17-s + ⋯
L(s)  = 1  + (0.147 + 0.691i)2-s + (−0.355 − 0.798i)3-s + (−0.456 + 0.203i)4-s + (−0.659 + 0.752i)5-s + (0.500 − 0.363i)6-s + (−0.0114 − 0.999i)7-s + (−0.207 − 0.286i)8-s + (0.157 − 0.175i)9-s + (−0.617 − 0.345i)10-s + (1.24 + 1.38i)11-s + (0.324 + 0.292i)12-s + (1.81 − 0.589i)13-s + (0.689 − 0.154i)14-s + (0.834 + 0.258i)15-s + (0.167 − 0.185i)16-s + (1.36 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.992 - 0.121i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.992 - 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24302 + 0.0756515i\)
\(L(\frac12)\) \(\approx\) \(1.24302 + 0.0756515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.207 - 0.978i)T \)
5 \( 1 + (1.47 - 1.68i)T \)
7 \( 1 + (0.0303 + 2.64i)T \)
good3 \( 1 + (0.615 + 1.38i)T + (-2.00 + 2.22i)T^{2} \)
11 \( 1 + (-4.14 - 4.60i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-6.53 + 2.12i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-5.62 + 0.591i)T + (16.6 - 3.53i)T^{2} \)
19 \( 1 + (5.14 + 2.29i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (0.103 + 0.485i)T + (-21.0 + 9.35i)T^{2} \)
29 \( 1 + (-1.23 - 0.899i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.00279 - 0.0266i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-3.67 - 3.31i)T + (3.86 + 36.7i)T^{2} \)
41 \( 1 + (1.32 + 4.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.08iT - 43T^{2} \)
47 \( 1 + (-0.0194 - 0.00204i)T + (45.9 + 9.77i)T^{2} \)
53 \( 1 + (-0.730 - 1.63i)T + (-35.4 + 39.3i)T^{2} \)
59 \( 1 + (0.215 + 0.0457i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-3.94 + 0.838i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (1.73 - 0.182i)T + (65.5 - 13.9i)T^{2} \)
71 \( 1 + (-1.06 - 0.770i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.66 + 5.09i)T + (7.63 - 72.6i)T^{2} \)
79 \( 1 + (0.468 - 4.45i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (6.33 + 8.71i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (8.43 - 1.79i)T + (81.3 - 36.1i)T^{2} \)
97 \( 1 + (-3.55 + 4.89i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.63541487540168789050342265301, −10.66393420502792222514483103902, −9.689248884446791128393528696280, −8.295820357774613766906796387590, −7.43086140073063702720850906350, −6.73770354262897348339459168190, −6.17986231936207791095957357654, −4.34022144268067697725770975412, −3.58722670986868364680453789699, −1.15601651172266109931936794884, 1.37476496127268856138798174677, 3.59021553108399224707146083050, 4.09717289747186590179209238156, 5.47550610503855698532625069289, 6.18149290186411312568559850786, 8.286192404147323119132207676871, 8.736542966309777871454638487674, 9.620789968757704555302958099232, 10.86090721351518765149595489451, 11.42013876777363590291057816384

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