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Properties

Label 2-350-35.9-c3-0-14
Degree $2$
Conductor $350$
Sign $0.890 + 0.455i$
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  + (−1.73 − i)2-s + (−0.866 + 0.5i)3-s + (1.99 + 3.46i)4-s + 1.99·6-s + (−16.4 − 8.5i)7-s − 7.99i·8-s + (−13 + 22.5i)9-s + (1 + 1.73i)11-s + (−3.46 − 1.99i)12-s − 8i·13-s + (19.9 + 31.1i)14-s + (−8 + 13.8i)16-s + (45.0 − 26i)17-s + (45.0 − 26i)18-s + (13 − 22.5i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.166 + 0.0962i)3-s + (0.249 + 0.433i)4-s + 0.136·6-s + (−0.888 − 0.458i)7-s − 0.353i·8-s + (−0.481 + 0.833i)9-s + (0.0274 + 0.0474i)11-s + (−0.0833 − 0.0481i)12-s − 0.170i·13-s + (0.381 + 0.595i)14-s + (−0.125 + 0.216i)16-s + (0.642 − 0.370i)17-s + (0.589 − 0.340i)18-s + (0.156 − 0.271i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9736376530\)
\(L(\frac12)\) \(\approx\) \(0.9736376530\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
5 \( 1 \)
7 \( 1 + (16.4 + 8.5i)T \)
good3 \( 1 + (0.866 - 0.5i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 8iT - 2.19e3T^{2} \)
17 \( 1 + (-45.0 + 26i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-13 + 22.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (58.0 + 33.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 69T + 2.43e4T^{2} \)
31 \( 1 + (-166 - 287. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-169. - 98i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 353T + 6.89e4T^{2} \)
43 \( 1 + 369iT - 7.95e4T^{2} \)
47 \( 1 + (-76.2 - 44i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-504. + 291i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (175 + 303. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-233.5 + 404. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (252. - 145.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 770T + 3.57e5T^{2} \)
73 \( 1 + (-543. + 314i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-585 + 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 525iT - 5.71e5T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 290iT - 9.12e5T^{2} \)
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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

−10.76299130111008238027096261808, −10.17372222537466648305008110469, −9.285322621731052966703755687279, −8.225486360739369178114549483094, −7.33500617520336423589650055899, −6.27643471778224662459782482851, −5.02432606006404905583500532490, −3.58680601034684426181497878349, −2.45390829331501117561722045075, −0.67203548155052435315907088086, 0.77090557588343301440811670050, 2.59602668472547866691495621827, 3.93930131743902559776971116223, 5.80069825471766918750732191831, 6.13870433384484715636002488979, 7.36326856470247681994340341011, 8.364122449151228956063283298459, 9.420164253845570032983137067710, 9.833009226583778584148197706312, 11.12900637379447286069801136069

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