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Properties

Label 2-350-25.14-c1-0-5
Degree $2$
Conductor $350$
Sign $-0.204 + 0.978i$
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 + (−3.01 − 0.979i)3-s + (−0.309 + 0.951i)4-s + (2.13 − 0.672i)5-s + (0.979 + 3.01i)6-s + i·7-s + (0.951 − 0.309i)8-s + (5.70 + 4.14i)9-s + (−1.79 − 1.33i)10-s + (4.60 − 3.34i)11-s + (1.86 − 2.56i)12-s + (−2.50 + 3.44i)13-s + (0.809 − 0.587i)14-s + (−7.08 − 0.0631i)15-s + (−0.809 − 0.587i)16-s + (2.28 − 0.740i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−1.74 − 0.565i)3-s + (−0.154 + 0.475i)4-s + (0.953 − 0.300i)5-s + (0.399 + 1.23i)6-s + 0.377i·7-s + (0.336 − 0.109i)8-s + (1.90 + 1.38i)9-s + (−0.568 − 0.420i)10-s + (1.38 − 1.00i)11-s + (0.537 − 0.740i)12-s + (−0.694 + 0.955i)13-s + (0.216 − 0.157i)14-s + (−1.83 − 0.0163i)15-s + (−0.202 − 0.146i)16-s + (0.553 − 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453191 - 0.557857i\)
\(L(\frac12)\) \(\approx\) \(0.453191 - 0.557857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 0.672i)T \)
7 \( 1 - iT \)
good3 \( 1 + (3.01 + 0.979i)T + (2.42 + 1.76i)T^{2} \)
11 \( 1 + (-4.60 + 3.34i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.50 - 3.44i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.28 + 0.740i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.63 + 5.03i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.84 + 3.91i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.328 - 1.01i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.02 - 3.16i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.02 + 8.28i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.30 + 3.85i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.49iT - 43T^{2} \)
47 \( 1 + (-6.98 - 2.27i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.36 - 2.39i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.24 + 4.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.54 + 1.84i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-7.18 + 2.33i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.97 - 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.55 + 3.51i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.75 - 5.39i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-8.31 + 2.70i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-9.09 + 6.61i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.33 + 0.434i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.31026595694557738329125240159, −10.55144738977695185623147476430, −9.502304527897529937708878727212, −8.737095372077286160098795004920, −7.06515960588602713788907064595, −6.36465667580191100223584424990, −5.47401550526431157786387620426, −4.36523803028135364828945857302, −2.13632996996568358881143223277, −0.828977024751787825615888453509, 1.36240737529507050605553947505, 4.06281577489695786220404541159, 5.16748061980794150514138643201, 6.02241349516355825827679953269, 6.63353167635856974621882761428, 7.69064370785809093680568851107, 9.507639264856641936943105151261, 9.983488390390988033973802517367, 10.47588058136331347389467328700, 11.69375366874940127051135092801

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