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Properties

Label 2-350-25.16-c1-0-1
Degree $2$
Conductor $350$
Sign $-0.668 - 0.743i$
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 + (1.01 + 3.12i)3-s + (0.309 + 0.951i)4-s + (−0.0943 + 2.23i)5-s + (1.01 − 3.12i)6-s − 7-s + (0.309 − 0.951i)8-s + (−6.30 + 4.58i)9-s + (1.38 − 1.75i)10-s + (2.00 + 1.45i)11-s + (−2.65 + 1.93i)12-s + (2.82 − 2.05i)13-s + (0.809 + 0.587i)14-s + (−7.07 + 1.97i)15-s + (−0.809 + 0.587i)16-s + (1.14 − 3.53i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.586 + 1.80i)3-s + (0.154 + 0.475i)4-s + (−0.0422 + 0.999i)5-s + (0.414 − 1.27i)6-s − 0.377·7-s + (0.109 − 0.336i)8-s + (−2.10 + 1.52i)9-s + (0.439 − 0.554i)10-s + (0.603 + 0.438i)11-s + (−0.767 + 0.557i)12-s + (0.784 − 0.569i)13-s + (0.216 + 0.157i)14-s + (−1.82 + 0.509i)15-s + (−0.202 + 0.146i)16-s + (0.278 − 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.668 - 0.743i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448311 + 1.00603i\)
\(L(\frac12)\) \(\approx\) \(0.448311 + 1.00603i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0943 - 2.23i)T \)
7 \( 1 + T \)
good3 \( 1 + (-1.01 - 3.12i)T + (-2.42 + 1.76i)T^{2} \)
11 \( 1 + (-2.00 - 1.45i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.82 + 2.05i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.14 + 3.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.409 + 1.25i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.30 + 3.12i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-3.17 - 9.78i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.999 - 3.07i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.08 + 5.87i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.85 + 1.35i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.26T + 43T^{2} \)
47 \( 1 + (-3.76 - 11.5i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.79 - 5.51i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.11 + 1.53i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.13 + 3.72i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.483 + 1.48i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.38 - 7.34i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.68 + 4.13i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.30 - 7.10i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.61 - 4.96i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.787 + 0.571i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.61 + 11.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.22014618330108295550847534505, −10.74675035766096967754246032930, −9.964989215075923858995083828673, −9.326976613668640545998665802087, −8.487071253588636068714223945678, −7.32251469497116731827150649727, −5.93654824747624400387609417533, −4.45466043198277044077566006622, −3.44676122867903046587276447698, −2.70632135475494124364396897839, 0.901167497221653461630233531220, 1.99310738060121317289477119694, 3.81835578953901013889852597852, 5.98218662395231778330407271838, 6.27892457981290969198730784040, 7.60914486107432608393496337373, 8.254959686887717003083350157295, 8.876607267242721273324405139340, 9.824469923188182041511844017656, 11.64511331849162667495192607421

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