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Properties

Label 2-350-25.9-c1-0-9
Degree $2$
Conductor $350$
Sign $0.955 - 0.295i$
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  + (−0.587 + 0.809i)2-s + (3.08 − 1.00i)3-s + (−0.309 − 0.951i)4-s + (1.12 + 1.93i)5-s + (−1.00 + 3.08i)6-s − i·7-s + (0.951 + 0.309i)8-s + (6.09 − 4.42i)9-s + (−2.22 − 0.226i)10-s + (−2.11 − 1.53i)11-s + (−1.90 − 2.62i)12-s + (1.40 + 1.92i)13-s + (0.809 + 0.587i)14-s + (5.40 + 4.83i)15-s + (−0.809 + 0.587i)16-s + (−6.80 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (1.78 − 0.578i)3-s + (−0.154 − 0.475i)4-s + (0.502 + 0.864i)5-s + (−0.409 + 1.25i)6-s − 0.377i·7-s + (0.336 + 0.109i)8-s + (2.03 − 1.47i)9-s + (−0.703 − 0.0715i)10-s + (−0.636 − 0.462i)11-s + (−0.550 − 0.757i)12-s + (0.388 + 0.534i)13-s + (0.216 + 0.157i)14-s + (1.39 + 1.24i)15-s + (−0.202 + 0.146i)16-s + (−1.65 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93004 + 0.291567i\)
\(L(\frac12)\) \(\approx\) \(1.93004 + 0.291567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-1.12 - 1.93i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-3.08 + 1.00i)T + (2.42 - 1.76i)T^{2} \)
11 \( 1 + (2.11 + 1.53i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.40 - 1.92i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.80 + 2.21i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.24 - 6.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.69 + 3.70i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.830 + 2.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.978 + 3.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.27 - 5.87i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.86 - 2.08i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.58iT - 43T^{2} \)
47 \( 1 + (6.84 - 2.22i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.57 - 0.836i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.59 + 6.97i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.723 - 0.526i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.998 + 0.324i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.940 + 2.89i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.68 - 9.19i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.82 + 11.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.21 - 2.01i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (9.66 + 7.01i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (11.0 - 3.58i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.31144749993075799355515062648, −10.22987049222655450104128337042, −9.493943925178395312671102983048, −8.521525566111867464204144891323, −7.915712306628079356473674658040, −6.88302982672634342598800243218, −6.29067645911187448128323942447, −4.25621374312239673418100835254, −2.93726269398254005404828106656, −1.86522503094403581965342037378, 1.93653036535563758149566268924, 2.79089525529366297385590526104, 4.16232604990969633921559284177, 5.07230159219755408164033389244, 7.06172895564438197604447429719, 8.284670292046002277027553871488, 8.811748807754492281009838735197, 9.334587575735974669035557732085, 10.26386556650586322368295135841, 11.12294549227878883638691125049

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