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Properties

Label 2-350-35.23-c2-0-20
Degree $2$
Conductor $350$
Sign $-0.993 - 0.114i$
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  + (−0.366 − 1.36i)2-s + (1.06 − 3.96i)3-s + (−1.73 + i)4-s − 5.80·6-s + (6.83 − 1.48i)7-s + (2 + 1.99i)8-s + (−6.81 − 3.93i)9-s + (−5.51 − 9.54i)11-s + (2.12 + 7.93i)12-s + (−11.4 − 11.4i)13-s + (−4.53 − 8.79i)14-s + (1.99 − 3.46i)16-s + (15.7 + 4.22i)17-s + (−2.88 + 10.7i)18-s + (−23.3 − 13.5i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.354 − 1.32i)3-s + (−0.433 + 0.250i)4-s − 0.968·6-s + (0.977 − 0.212i)7-s + (0.250 + 0.249i)8-s + (−0.757 − 0.437i)9-s + (−0.501 − 0.868i)11-s + (0.177 + 0.661i)12-s + (−0.877 − 0.877i)13-s + (−0.324 − 0.628i)14-s + (0.124 − 0.216i)16-s + (0.926 + 0.248i)17-s + (−0.160 + 0.597i)18-s + (−1.23 − 0.710i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.993 - 0.114i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.993 - 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0827540 + 1.43976i\)
\(L(\frac12)\) \(\approx\) \(0.0827540 + 1.43976i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-6.83 + 1.48i)T \)
good3 \( 1 + (-1.06 + 3.96i)T + (-7.79 - 4.5i)T^{2} \)
11 \( 1 + (5.51 + 9.54i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (11.4 + 11.4i)T + 169iT^{2} \)
17 \( 1 + (-15.7 - 4.22i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (23.3 + 13.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (20.9 - 5.61i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 13.2iT - 841T^{2} \)
31 \( 1 + (-24.6 - 42.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.12 + 15.3i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 23.0T + 1.68e3T^{2} \)
43 \( 1 + (-35.1 - 35.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-13.6 - 51.1i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-15.6 + 58.3i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (13.7 - 7.96i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-16.9 + 29.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-84.0 - 22.5i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 123.T + 5.04e3T^{2} \)
73 \( 1 + (-28.4 + 106. i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (109. + 63.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (6.82 + 6.82i)T + 6.88e3iT^{2} \)
89 \( 1 + (-49.9 - 28.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-29.2 + 29.2i)T - 9.40e3iT^{2} \)
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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

−10.84569706366067070724012495296, −10.09510694685808621307282567336, −8.569737125125519151634636983560, −8.055839884574720188500691365665, −7.33200921182168878033811514112, −5.94186677194063472256454571645, −4.69162884391002624083456486589, −3.03509218132403290118748586557, −1.97126035102247110604004305139, −0.67424128065431592907823678080, 2.20000339595977187642254244524, 4.10244256028976271507117248016, 4.68767452690474800820462635947, 5.69177752758431921529032850940, 7.18410328966199329634667786594, 8.120586250874783081776536535911, 8.946789823625778376928725402915, 9.995848230729430061836616463915, 10.30328740408597371154482561880, 11.65824855654196747935681265278

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