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Properties

Label 2-350-35.24-c2-0-8
Degree $2$
Conductor $350$
Sign $0.927 - 0.374i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.765881 + 0.148951i\)
\(L(\frac12)\) \(\approx\) \(0.765881 + 0.148951i\)
\(L(2)\) 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

−11.08109347776999743623508720958, −10.31647004299700140680143112532, −9.518739509362285191194078731999, −8.888143712180021507621185229567, −7.52234344733391313413972521135, −6.29628346518633059334018131242, −5.66594407896339443561376225548, −4.32324009006454798552347896956, −3.21091147338346473225153187818, −0.62805907411968647615959529145, 1.01227755775309271203189754409, 2.28820112531440924315362471350, 3.89592591665607335547653924097, 5.73515535453267331960535839129, 6.58733264269990239377190685600, 7.27052699365332947399336163106, 8.355336305080321668578407361733, 9.491215564431651194217822144089, 10.22288525232290166474716632342, 11.50594528558110860073379755395

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