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Properties

Label 2-350-35.34-c4-0-0
Degree $2$
Conductor $350$
Sign $-0.781 - 0.623i$
Analytic cond. $36.1794$
Root an. cond. $6.01493$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s − 11.2·3-s − 8.00·4-s + 31.8i·6-s + (20.6 + 44.4i)7-s + 22.6i·8-s + 46.0·9-s − 11.8·11-s + 90.1·12-s + 20.6·13-s + (125. − 58.2i)14-s + 64.0·16-s + 289.·17-s − 130. i·18-s − 104. i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.25·3-s − 0.500·4-s + 0.885i·6-s + (0.420 + 0.907i)7-s + 0.353i·8-s + 0.568·9-s − 0.0977·11-s + 0.626·12-s + 0.121·13-s + (0.641 − 0.297i)14-s + 0.250·16-s + 1.00·17-s − 0.401i·18-s − 0.289i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.781 - 0.623i$
Analytic conductor: \(36.1794\)
Root analytic conductor: \(6.01493\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :2),\ -0.781 - 0.623i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.08689852071\)
\(L(\frac12)\) \(\approx\) \(0.08689852071\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
5 \( 1 \)
7 \( 1 + (-20.6 - 44.4i)T \)
good3 \( 1 + 11.2T + 81T^{2} \)
11 \( 1 + 11.8T + 1.46e4T^{2} \)
13 \( 1 - 20.6T + 2.85e4T^{2} \)
17 \( 1 - 289.T + 8.35e4T^{2} \)
19 \( 1 + 104. iT - 1.30e5T^{2} \)
23 \( 1 - 73.5iT - 2.79e5T^{2} \)
29 \( 1 + 950.T + 7.07e5T^{2} \)
31 \( 1 + 1.38e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.27e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + 96.2iT - 3.41e6T^{2} \)
47 \( 1 + 186.T + 4.87e6T^{2} \)
53 \( 1 - 4.37e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.65e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.20e3iT - 1.38e7T^{2} \)
67 \( 1 + 552. iT - 2.01e7T^{2} \)
71 \( 1 + 8.48e3T + 2.54e7T^{2} \)
73 \( 1 - 317.T + 2.83e7T^{2} \)
79 \( 1 - 624.T + 3.89e7T^{2} \)
83 \( 1 + 7.66e3T + 4.74e7T^{2} \)
89 \( 1 + 4.19e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.29e4T + 8.85e7T^{2} \)
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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

−11.39037728371373701001624268098, −10.55964631112440737255567279703, −9.593703447449556360993464808539, −8.596147242610507200484350405348, −7.44108461717625216519306701506, −5.92652748092475894953906541154, −5.46857108252337917072275357509, −4.32807161142433165180827338568, −2.79144840505209130692477597692, −1.35954299808308929008308289622, 0.03512007377958707304507088347, 1.25951631873138198500016754920, 3.62106583496691192548617198816, 4.85299547339568825687486533027, 5.57691105343223588692124069032, 6.58213079379436661536278531499, 7.43815929920493059343987694965, 8.371489612060574454385843130040, 9.728717745129484919234115922200, 10.58890817565774371290380818784

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