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Properties

Label 2-350-35.4-c1-0-11
Degree $2$
Conductor $350$
Sign $-0.657 + 0.753i$
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  + (0.866 − 0.5i)2-s + (−1.73 − i)3-s + (0.499 − 0.866i)4-s − 1.99·6-s + (1.73 − 2i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s + (−1.73 + 0.999i)12-s − 5i·13-s + (0.499 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−5.19 − 3i)17-s + (0.866 + 0.499i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.999 − 0.577i)3-s + (0.249 − 0.433i)4-s − 0.816·6-s + (0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s + (−0.499 + 0.288i)12-s − 1.38i·13-s + (0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (−1.26 − 0.727i)17-s + (0.204 + 0.117i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.657 + 0.753i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504980 - 1.11037i\)
\(L(\frac12)\) \(\approx\) \(0.504980 - 1.11037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.52 + 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.59 - 1.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−11.21426584324179889035389135531, −10.66432525565130644945385743177, −9.640834992686618269549291368786, −8.011349811083964745486437480054, −7.16317915817777626031155599038, −6.17272196958069003853658844294, −5.15131083899429726382587808062, −4.29458182247929823610823550982, −2.52675734658597325433720398873, −0.789436370322648087206334326454, 2.33743370097526161974683920623, 4.19430697596854804586292255840, 4.89865971017124170323198390837, 5.93435305070075924513073593131, 6.57523811281186947766622127201, 8.153479303787426702305880534824, 8.862105941442708361593986089738, 10.28590181227349608224714666112, 11.20376505006161369898822958249, 11.64161348839120777532506698775

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