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Properties

Label 2-350-25.11-c1-0-2
Degree $2$
Conductor $350$
Sign $-0.228 - 0.973i$
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  + (−0.809 + 0.587i)2-s + (−0.413 + 1.27i)3-s + (0.309 − 0.951i)4-s + (−1.11 − 1.93i)5-s + (−0.413 − 1.27i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.978 + 0.710i)9-s + (2.04 + 0.909i)10-s + (−3.33 + 2.42i)11-s + (1.08 + 0.786i)12-s + (3.59 + 2.61i)13-s + (−0.809 + 0.587i)14-s + (2.92 − 0.622i)15-s + (−0.809 − 0.587i)16-s + (2.06 + 6.35i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.238 + 0.734i)3-s + (0.154 − 0.475i)4-s + (−0.499 − 0.866i)5-s + (−0.168 − 0.519i)6-s + 0.377·7-s + (0.109 + 0.336i)8-s + (0.326 + 0.236i)9-s + (0.645 + 0.287i)10-s + (−1.00 + 0.730i)11-s + (0.312 + 0.227i)12-s + (0.997 + 0.724i)13-s + (−0.216 + 0.157i)14-s + (0.755 − 0.160i)15-s + (−0.202 − 0.146i)16-s + (0.500 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513607 + 0.648011i\)
\(L(\frac12)\) \(\approx\) \(0.513607 + 0.648011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (1.11 + 1.93i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.413 - 1.27i)T + (-2.42 - 1.76i)T^{2} \)
11 \( 1 + (3.33 - 2.42i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.59 - 2.61i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.06 - 6.35i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.290 + 0.895i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.44 + 1.77i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.64 - 8.14i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.36 + 7.28i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.85 - 6.43i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.23 + 3.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.11T + 43T^{2} \)
47 \( 1 + (-0.997 + 3.07i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.59 - 4.89i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.03 - 5.11i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.77 + 4.19i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.10 - 6.49i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.73 + 11.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.76 - 3.46i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.96 - 6.03i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.15 + 9.71i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.43 + 6.85i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.62 + 17.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.42202504541480162558255245939, −10.71776651842613051064015868011, −9.896260332782703340247212221593, −8.884924152937601690844573734781, −8.129699089107815275101090089413, −7.24088748140427026294909341151, −5.77335415312018033522280121970, −4.83088394766714189238539083840, −3.96792861848663270653090232949, −1.62491225269530554402890918999, 0.76833602436388745731555009598, 2.61020050910782782645352544110, 3.69621898626647664512435786227, 5.49588056067814864356559608944, 6.66351172221132784699114581521, 7.65610075796399594653049707978, 8.088745439934142376528699762948, 9.486492300190334590790600147795, 10.51727883094548453160731451645, 11.25427272639157725614839325851

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