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Properties

Label 2-350-35.17-c1-0-3
Degree $2$
Conductor $350$
Sign $0.133 - 0.991i$
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  + (−0.258 + 0.965i)2-s + (0.707 − 0.189i)3-s + (−0.866 − 0.499i)4-s + 0.732i·6-s + (1.48 + 2.19i)7-s + (0.707 − 0.707i)8-s + (−2.13 + 1.23i)9-s + (0.633 − 1.09i)11-s + (−0.707 − 0.189i)12-s + (4.38 + 4.38i)13-s + (−2.49 + 0.866i)14-s + (0.500 + 0.866i)16-s + (0.120 + 0.448i)17-s + (−0.637 − 2.38i)18-s + (1.46 + 1.26i)21-s + (0.896 + 0.896i)22-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.408 − 0.109i)3-s + (−0.433 − 0.249i)4-s + 0.298i·6-s + (0.560 + 0.827i)7-s + (0.249 − 0.249i)8-s + (−0.711 + 0.410i)9-s + (0.191 − 0.331i)11-s + (−0.204 − 0.0546i)12-s + (1.21 + 1.21i)13-s + (−0.668 + 0.231i)14-s + (0.125 + 0.216i)16-s + (0.0291 + 0.108i)17-s + (−0.150 − 0.561i)18-s + (0.319 + 0.276i)21-s + (0.191 + 0.191i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.133 - 0.991i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.133 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02701 + 0.897785i\)
\(L(\frac12)\) \(\approx\) \(1.02701 + 0.897785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-1.48 - 2.19i)T \)
good3 \( 1 + (-0.707 + 0.189i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.38 - 4.38i)T + 13iT^{2} \)
17 \( 1 + (-0.120 - 0.448i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.01 - 1.34i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.79 + 6.69i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + (7.58 - 7.58i)T - 43iT^{2} \)
47 \( 1 + (3.79 + 1.01i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.12 + 7.91i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-7.09 + 12.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.29 - 3.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.67 + 0.984i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (7.72 - 2.07i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.86 + 5.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \)
89 \( 1 + (5.59 + 9.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.05 + 8.05i)T - 97iT^{2} \)
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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

−11.42552124544085577474924027939, −11.02167554503612539745483172453, −9.382661189333795856089301118355, −8.743613942283881978262923012749, −8.180446433719655747873170947979, −6.94404894809886638400434605387, −5.91330381661320268799603560397, −5.00063906822068349958512685125, −3.51631470276715259025766633912, −1.86001097228366366674254480452, 1.10140747924229547525888498639, 2.92013669177662188011472420882, 3.86307213881589815981161949513, 5.12286967838414005001115508427, 6.48989740458129946274884706269, 7.890678177314281105885791261420, 8.471208596451988159595549771923, 9.500451027532271178032367988697, 10.48517202425614638602746912012, 11.15860304557376902091879478951

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