Properties

Label 2-350-35.19-c2-0-14
Degree $2$
Conductor $350$
Sign $-0.256 - 0.966i$
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  + (1.22 + 0.707i)2-s + (2.09 + 3.62i)3-s + (0.999 + 1.73i)4-s + 5.91i·6-s + (6.63 − 2.24i)7-s + 2.82i·8-s + (−4.24 + 7.34i)9-s + (6.62 + 11.4i)11-s + (−4.18 + 7.24i)12-s − 5.49·13-s + (9.70 + 1.94i)14-s + (−2.00 + 3.46i)16-s + (−6.77 − 11.7i)17-s + (−10.3 + 6.00i)18-s + (0.621 + 0.358i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.696 + 1.20i)3-s + (0.249 + 0.433i)4-s + 0.985i·6-s + (0.947 − 0.320i)7-s + 0.353i·8-s + (−0.471 + 0.816i)9-s + (0.601 + 1.04i)11-s + (−0.348 + 0.603i)12-s − 0.422·13-s + (0.693 + 0.138i)14-s + (−0.125 + 0.216i)16-s + (−0.398 − 0.690i)17-s + (−0.577 + 0.333i)18-s + (0.0327 + 0.0188i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.256 - 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.94146 + 2.52370i\)
\(L(\frac12)\) \(\approx\) \(1.94146 + 2.52370i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-6.63 + 2.24i)T \)
good3 \( 1 + (-2.09 - 3.62i)T + (-4.5 + 7.79i)T^{2} \)
11 \( 1 + (-6.62 - 11.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 5.49T + 169T^{2} \)
17 \( 1 + (6.77 + 11.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.621 - 0.358i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (1.96 + 1.13i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 20.4T + 841T^{2} \)
31 \( 1 + (-21.3 + 12.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (56.2 + 32.4i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 21.0iT - 1.68e3T^{2} \)
43 \( 1 + 6.48iT - 1.84e3T^{2} \)
47 \( 1 + (23.8 - 41.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (19.0 - 11.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-72.5 + 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-57.3 - 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-80.2 + 46.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (65.4 + 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (38.1 - 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 107.T + 6.88e3T^{2} \)
89 \( 1 + (-145. - 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 25.5T + 9.40e3T^{2} \)
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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

−11.53713729259319417305223162634, −10.55655992427776098707169036880, −9.626243961402678765182596192038, −8.831937789314481586710791107716, −7.74367928229524370517167507990, −6.84383142695037501924209301074, −5.19496760381620058989370695728, −4.50559119289496819163671925701, −3.68602006274703917896851142318, −2.18967846185389159997982615713, 1.29420689000514094934392101384, 2.30807674049925710909305415363, 3.58077388296765750031845298279, 5.01961280679001427881559291346, 6.20587806680973636910182404612, 7.13431136358842773205913552557, 8.266689169454806246965451748807, 8.773363453824541642366862501950, 10.25643259546221906753378288061, 11.41761409136762686299641508422

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