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Properties

Label 2-350-35.2-c2-0-1
Degree $2$
Conductor $350$
Sign $-0.0214 - 0.999i$
Analytic cond. $9.53680$
Root an. cond. $3.08817$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (−2.63 − 0.707i)3-s + (1.73 − i)4-s − 3.86·6-s + (−6.78 + 1.73i)7-s + (1.99 − 2i)8-s + (−1.33 − 0.768i)9-s + (10.0 + 17.3i)11-s + (−5.27 + 1.41i)12-s + (−2.21 + 2.21i)13-s + (−8.63 + 4.84i)14-s + (1.99 − 3.46i)16-s + (−3.19 + 11.9i)17-s + (−2.09 − 0.562i)18-s + (−15.7 − 9.11i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.879 − 0.235i)3-s + (0.433 − 0.250i)4-s − 0.643·6-s + (−0.968 + 0.247i)7-s + (0.249 − 0.250i)8-s + (−0.147 − 0.0853i)9-s + (0.911 + 1.57i)11-s + (−0.439 + 0.117i)12-s + (−0.170 + 0.170i)13-s + (−0.616 + 0.346i)14-s + (0.124 − 0.216i)16-s + (−0.188 + 0.701i)17-s + (−0.116 − 0.0312i)18-s + (−0.831 − 0.479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0214 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0214 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.0214 - 0.999i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.0214 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.699581 + 0.714766i\)
\(L(\frac12)\) \(\approx\) \(0.699581 + 0.714766i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 \)
7 \( 1 + (6.78 - 1.73i)T \)
good3 \( 1 + (2.63 + 0.707i)T + (7.79 + 4.5i)T^{2} \)
11 \( 1 + (-10.0 - 17.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (2.21 - 2.21i)T - 169iT^{2} \)
17 \( 1 + (3.19 - 11.9i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (15.7 + 9.11i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.6 - 39.7i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 19.0iT - 841T^{2} \)
31 \( 1 + (3.14 + 5.43i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (36.6 - 9.82i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 16.3T + 1.68e3T^{2} \)
43 \( 1 + (-4.26 + 4.26i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.9 + 3.19i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (51.7 + 13.8i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (82.3 - 47.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (4.22 - 7.32i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.94 - 10.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 94.8T + 5.04e3T^{2} \)
73 \( 1 + (7.87 + 2.11i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-15.1 - 8.72i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-63.0 + 63.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (67.0 + 38.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (105. + 105. i)T + 9.40e3iT^{2} \)
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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

−11.73702264627463011678826261296, −10.82459153056269895143030613657, −9.782009319492285048871348243822, −8.965010353454210610786264878516, −7.15050648807357540351686582358, −6.60765787731684628222923620397, −5.68174682514069002301256859816, −4.57853600658655583742737560492, −3.36909372708727100946384008119, −1.73156284992252938926146584433, 0.39671176946970811038469733563, 2.86088903464508881917130250408, 3.99432068105339751161487024489, 5.18130859939122898577036141135, 6.28247950306991147112049967737, 6.57232618978924773220170553554, 8.188481928859547881512165932956, 9.186580393285526385519874762811, 10.53113502455623041867665732970, 11.03705572954680894699186030756

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