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Properties

Label 2-350-7.4-c3-0-2
Degree $2$
Conductor $350$
Sign $-0.00887 + 0.999i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−4.59 + 7.95i)3-s + (−1.99 + 3.46i)4-s + 18.3·6-s + (−8.37 + 16.5i)7-s + 7.99·8-s + (−28.7 − 49.7i)9-s + (−17.7 + 30.7i)11-s + (−18.3 − 31.8i)12-s − 45.5·13-s + (36.9 − 2.01i)14-s + (−8 − 13.8i)16-s + (−46.8 + 81.1i)17-s + (−57.4 + 99.4i)18-s + (−22.4 − 38.8i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.884 + 1.53i)3-s + (−0.249 + 0.433i)4-s + 1.25·6-s + (−0.452 + 0.891i)7-s + 0.353·8-s + (−1.06 − 1.84i)9-s + (−0.486 + 0.842i)11-s + (−0.442 − 0.765i)12-s − 0.971·13-s + (0.706 − 0.0385i)14-s + (−0.125 − 0.216i)16-s + (−0.668 + 1.15i)17-s + (−0.751 + 1.30i)18-s + (−0.270 − 0.469i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00887 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00887 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.00887 + 0.999i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.00887 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08518606125\)
\(L(\frac12)\) \(\approx\) \(0.08518606125\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
5 \( 1 \)
7 \( 1 + (8.37 - 16.5i)T \)
good3 \( 1 + (4.59 - 7.95i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (17.7 - 30.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 45.5T + 2.19e3T^{2} \)
17 \( 1 + (46.8 - 81.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (22.4 + 38.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-61.2 - 106. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 17.8T + 2.43e4T^{2} \)
31 \( 1 + (13.5 - 23.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-39.0 - 67.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 21.0T + 6.89e4T^{2} \)
43 \( 1 - 467.T + 7.95e4T^{2} \)
47 \( 1 + (289. + 501. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-80.6 + 139. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-279. + 484. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-54.3 - 94.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (203. - 353. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.15e3T + 3.57e5T^{2} \)
73 \( 1 + (128. - 222. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-426. - 739. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 828.T + 5.71e5T^{2} \)
89 \( 1 + (82.4 + 142. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 38.6T + 9.12e5T^{2} \)
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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

−11.49369594297330726515791138658, −10.76478266948096831248445103249, −9.873484690971867315360397414723, −9.455975044821418707419923857105, −8.456824512925687606710426482798, −6.86191004542168951339125415637, −5.55990897775982291947740756060, −4.79202611064632316831076633811, −3.71952168431944701761057809738, −2.36957089075263574098848151012, 0.05117631649168957303892251938, 0.854153894498952694955603928820, 2.56356972314046997204514208783, 4.70112590706780797405643056930, 5.81827572642937841972256757592, 6.66888862397614929558735372987, 7.34056329142440626512530901507, 8.030540435294665896461852147267, 9.320939671717160073872494601324, 10.59475774034243690680674253314

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