Properties

Label 2-350-25.4-c1-0-4
Degree $2$
Conductor $350$
Sign $0.715 - 0.698i$
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  + (−0.951 − 0.309i)2-s + (0.310 − 0.427i)3-s + (0.809 + 0.587i)4-s + (1.73 + 1.40i)5-s + (−0.427 + 0.310i)6-s + i·7-s + (−0.587 − 0.809i)8-s + (0.840 + 2.58i)9-s + (−1.21 − 1.87i)10-s + (−0.793 + 2.44i)11-s + (0.502 − 0.163i)12-s + (−3.01 + 0.980i)13-s + (0.309 − 0.951i)14-s + (1.14 − 0.304i)15-s + (0.309 + 0.951i)16-s + (0.00697 + 0.00959i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.179 − 0.246i)3-s + (0.404 + 0.293i)4-s + (0.776 + 0.629i)5-s + (−0.174 + 0.126i)6-s + 0.377i·7-s + (−0.207 − 0.286i)8-s + (0.280 + 0.862i)9-s + (−0.384 − 0.593i)10-s + (−0.239 + 0.736i)11-s + (0.144 − 0.0470i)12-s + (−0.836 + 0.271i)13-s + (0.0825 − 0.254i)14-s + (0.294 − 0.0787i)15-s + (0.0772 + 0.237i)16-s + (0.00169 + 0.00232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.715 - 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03675 + 0.422444i\)
\(L(\frac12)\) \(\approx\) \(1.03675 + 0.422444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-1.73 - 1.40i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.310 + 0.427i)T + (-0.927 - 2.85i)T^{2} \)
11 \( 1 + (0.793 - 2.44i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.01 - 0.980i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.00697 - 0.00959i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.938 + 0.681i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.24 + 0.728i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.76 - 1.28i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.70 + 1.96i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-10.1 + 3.31i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.05 - 6.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + (-5.05 + 6.95i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.39 - 8.79i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.71 - 8.35i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.61 + 11.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (4.33 + 5.96i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.26 - 1.64i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.83 + 2.21i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.57 + 6.95i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.48 + 8.93i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.71 + 8.35i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.17 + 4.37i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.46139974073814201510920515278, −10.41563116292553339927731268243, −9.887396078925255621644839953698, −8.968193384838845741748572633518, −7.72521721696976803134192807222, −7.11632908353350458761974814419, −5.93828572832955704377208286736, −4.64745564566487239479782565919, −2.71125230479522988807556619277, −1.95764477859127785232635737020, 0.995204332672461398064398024978, 2.78216462639570463819015115987, 4.40048132912103495178290169394, 5.66965183247181380480841549102, 6.55397331691379745547258133071, 7.79630707781976549851874934076, 8.664419336245798541448823758201, 9.701578846901439826195685604192, 9.979349545988155512159436176973, 11.20399515019961981357132857537

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