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Properties

Label 2-350-35.4-c1-0-9
Degree $2$
Conductor $350$
Sign $0.943 + 0.330i$
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 + (0.866 − 2.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.73 − 0.999i)12-s + 2i·13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2.59 − 1.5i)17-s + (0.866 + 0.499i)18-s + (4 + 6.92i)19-s + (4 − 3.46i)21-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 + (0.327 − 0.944i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.499 − 0.288i)12-s + 0.554i·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.630 − 0.363i)17-s + (0.204 + 0.117i)18-s + (0.917 + 1.58i)19-s + (0.872 − 0.755i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.943 + 0.330i$
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: \((2,\ 350,\ (\ :1/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44457 - 0.415724i\)
\(L(\frac12)\) \(\approx\) \(2.44457 - 0.415724i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 2.5i)T \)
good3 \( 1 + (-1.73 - i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.79 - 4.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−11.56639737155773847778700222764, −10.27296779392870515822522957949, −9.890765753257729208540769658926, −8.699911305977165600424963121196, −7.76109591213083851158909908024, −6.62978360500037697049753109830, −5.21924846825589436288397845283, −4.00720248780130839488293762738, −3.44811652135354053695252044191, −1.83337462562899330620533326842, 2.17517875351617811351186055467, 3.01938248301581628065125983446, 4.56838691321894256699761593191, 5.69251669405258730205831880288, 6.78881129668228894725194555255, 7.86956297253630029962147387995, 8.482366566024908133034104213175, 9.342047141479144464469525682380, 10.78960325418306708539198359353, 11.82297371857778738686245418375

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