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Properties

Label 2-350-35.33-c1-0-7
Degree $2$
Conductor $350$
Sign $0.0481 - 0.998i$
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  + (0.258 + 0.965i)2-s + (2.80 + 0.752i)3-s + (−0.866 + 0.499i)4-s + 2.90i·6-s + (−0.559 + 2.58i)7-s + (−0.707 − 0.707i)8-s + (4.71 + 2.72i)9-s + (−1.83 − 3.17i)11-s + (−2.80 + 0.752i)12-s + (−0.830 + 0.830i)13-s + (−2.64 + 0.128i)14-s + (0.500 − 0.866i)16-s + (0.204 − 0.761i)17-s + (−1.41 + 5.26i)18-s + (1.09 − 1.89i)19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (1.62 + 0.434i)3-s + (−0.433 + 0.249i)4-s + 1.18i·6-s + (−0.211 + 0.977i)7-s + (−0.249 − 0.249i)8-s + (1.57 + 0.908i)9-s + (−0.553 − 0.958i)11-s + (−0.810 + 0.217i)12-s + (−0.230 + 0.230i)13-s + (−0.706 + 0.0343i)14-s + (0.125 − 0.216i)16-s + (0.0494 − 0.184i)17-s + (−0.332 + 1.24i)18-s + (0.251 − 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0481 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0481 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.0481 - 0.998i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.0481 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59254 + 1.51757i\)
\(L(\frac12)\) \(\approx\) \(1.59254 + 1.51757i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.559 - 2.58i)T \)
good3 \( 1 + (-2.80 - 0.752i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.83 + 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.830 - 0.830i)T - 13iT^{2} \)
17 \( 1 + (-0.204 + 0.761i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.09 + 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.54 + 1.21i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (-0.0359 + 0.0207i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0664 + 0.248i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (-0.474 - 0.474i)T + 43iT^{2} \)
47 \( 1 + (6.18 - 1.65i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.04 - 7.64i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.35 - 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 0.996i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.39 + 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (9.52 + 2.55i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.6 + 6.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.73 + 9.73i)T - 83iT^{2} \)
89 \( 1 + (-0.715 + 1.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \)
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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

−11.85787190229967680718110526203, −10.52294105470042550815795608105, −9.355516541990828992277212203613, −8.886486889590210205371262386642, −8.137619004486867947908994669666, −7.18843369670567561353386372082, −5.83123998114139475190422020131, −4.70414664190711656970853144445, −3.35727660334058257620337645509, −2.54100691588947425502230988367, 1.54953042068715610371355693672, 2.83650373226989320272549799210, 3.73253940700257199117181679147, 4.90866456067868718547425310357, 6.82561617882059155182268610526, 7.64157921743178522098266581483, 8.442744400129272707317691624149, 9.648465404677998620132714804572, 10.03406692169366521233390710483, 11.21542501062026340369787795628

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