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Properties

Label 2-350-35.3-c1-0-11
Degree $2$
Conductor $350$
Sign $0.139 + 0.990i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.523 − 1.95i)3-s + (0.866 + 0.499i)4-s − 2.02i·6-s + (−1.83 − 1.90i)7-s + (0.707 + 0.707i)8-s + (−0.941 + 0.543i)9-s + (2.01 − 3.49i)11-s + (0.523 − 1.95i)12-s + (−0.204 + 0.204i)13-s + (−1.28 − 2.31i)14-s + (0.500 + 0.866i)16-s + (1.97 − 0.527i)17-s + (−1.05 + 0.281i)18-s + (−3.10 − 5.37i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.302 − 1.12i)3-s + (0.433 + 0.249i)4-s − 0.825i·6-s + (−0.695 − 0.718i)7-s + (0.249 + 0.249i)8-s + (−0.313 + 0.181i)9-s + (0.609 − 1.05i)11-s + (0.151 − 0.563i)12-s + (−0.0568 + 0.0568i)13-s + (−0.343 − 0.618i)14-s + (0.125 + 0.216i)16-s + (0.477 − 0.128i)17-s + (−0.247 + 0.0663i)18-s + (−0.711 − 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.139 + 0.990i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27599 - 1.10860i\)
\(L(\frac12)\) \(\approx\) \(1.27599 - 1.10860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (1.83 + 1.90i)T \)
good3 \( 1 + (0.523 + 1.95i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.01 + 3.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.204 - 0.204i)T - 13iT^{2} \)
17 \( 1 + (-1.97 + 0.527i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.10 + 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.17 - 4.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 + (-6.33 - 3.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.46 - 1.19i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 + (-4.97 - 4.97i)T + 43iT^{2} \)
47 \( 1 + (-0.0815 + 0.304i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.00 + 2.14i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.427 + 0.740i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.99 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.817 - 3.05i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + (-2.98 - 11.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.39 - 2.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.85 + 3.85i)T - 83iT^{2} \)
89 \( 1 + (1.53 + 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \)
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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

−11.53127731060789354577411460903, −10.67245387771782947769515883201, −9.380625664225827483194797056332, −8.165632412090985988043208392766, −7.04743996612794565745226527664, −6.60727590549547280591861506112, −5.63064977335208255696733558829, −4.13958353039875854454957545243, −2.93983128496779364962434032182, −1.06492513120141071438471077709, 2.31974507946697229372340349844, 3.84610160417860417971734271056, 4.50854666747625040261683452207, 5.73062473515201393422058564548, 6.48609511886192330102366902263, 7.947686008707766602134839869146, 9.359001809995328345326387224252, 9.958601267894662461643188944580, 10.65147393941608991226732361296, 11.93643194202947465747211398618

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