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Properties

Label 2-350-25.11-c1-0-7
Degree $2$
Conductor $350$
Sign $0.686 + 0.727i$
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 + (−0.329 + 1.01i)3-s + (0.309 − 0.951i)4-s + (−1.58 − 1.57i)5-s + (0.329 + 1.01i)6-s + 7-s + (−0.309 − 0.951i)8-s + (1.50 + 1.09i)9-s + (−2.20 − 0.347i)10-s + (5.13 − 3.72i)11-s + (0.862 + 0.626i)12-s + (1.45 + 1.05i)13-s + (0.809 − 0.587i)14-s + (2.12 − 1.08i)15-s + (−0.809 − 0.587i)16-s + (−2.37 − 7.31i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.190 + 0.585i)3-s + (0.154 − 0.475i)4-s + (−0.707 − 0.706i)5-s + (0.134 + 0.414i)6-s + 0.377·7-s + (−0.109 − 0.336i)8-s + (0.502 + 0.364i)9-s + (−0.698 − 0.109i)10-s + (1.54 − 1.12i)11-s + (0.249 + 0.180i)12-s + (0.404 + 0.293i)13-s + (0.216 − 0.157i)14-s + (0.548 − 0.280i)15-s + (−0.202 − 0.146i)16-s + (−0.576 − 1.77i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.686 + 0.727i$
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: \((2,\ 350,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57769 - 0.680572i\)
\(L(\frac12)\) \(\approx\) \(1.57769 - 0.680572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 1.57i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.329 - 1.01i)T + (-2.42 - 1.76i)T^{2} \)
11 \( 1 + (-5.13 + 3.72i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.45 - 1.05i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.37 + 7.31i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.333 - 1.02i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.576 + 0.418i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.41 - 4.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.218 + 0.673i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.30 - 3.85i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.50 - 2.54i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (0.625 - 1.92i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.18 - 6.71i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.46 - 1.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.96 - 6.51i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.36 + 13.4i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.607 - 1.87i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.30 - 0.950i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.74 + 8.43i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.24 - 3.82i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.82 + 7.13i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.15 - 6.63i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.50661035930707323858391987868, −10.81163429304974794246594553184, −9.423281597672593836318609250363, −8.886757146984282066211387504560, −7.56214082624305910615220084663, −6.35402254991009802082152260285, −5.02308197720036514944273450401, −4.36302246529871917720068176017, −3.37076115976645655730249619258, −1.25795126677635860771150893448, 1.80565636416929525038108301529, 3.75146850679793108537853836369, 4.35293013496349536225004849774, 6.14204690814753760292615487870, 6.73814874628400993675384437330, 7.53033094391669263193265098704, 8.508829332274553352096208320710, 9.801895329912445637944235464389, 11.01554981553509007732394729820, 11.78578518484203505877442649390

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