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Properties

Label 2-350-175.33-c1-0-5
Degree $2$
Conductor $350$
Sign $0.851 - 0.523i$
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  + (0.838 − 0.544i)2-s + (−0.977 + 0.375i)3-s + (0.406 − 0.913i)4-s + (1.20 + 1.88i)5-s + (−0.615 + 0.847i)6-s + (2.05 + 1.66i)7-s + (−0.156 − 0.987i)8-s + (−1.41 + 1.27i)9-s + (2.03 + 0.920i)10-s + (−1.84 + 2.04i)11-s + (−0.0548 + 1.04i)12-s + (0.551 + 1.08i)13-s + (2.63 + 0.281i)14-s + (−1.88 − 1.38i)15-s + (−0.669 − 0.743i)16-s + (0.714 − 0.578i)17-s + ⋯
L(s)  = 1  + (0.593 − 0.385i)2-s + (−0.564 + 0.216i)3-s + (0.203 − 0.456i)4-s + (0.539 + 0.841i)5-s + (−0.251 + 0.345i)6-s + (0.775 + 0.630i)7-s + (−0.0553 − 0.349i)8-s + (−0.471 + 0.424i)9-s + (0.644 + 0.291i)10-s + (−0.556 + 0.617i)11-s + (−0.0158 + 0.301i)12-s + (0.153 + 0.300i)13-s + (0.703 + 0.0752i)14-s + (−0.487 − 0.358i)15-s + (−0.167 − 0.185i)16-s + (0.173 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.851 - 0.523i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.851 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63385 + 0.462126i\)
\(L(\frac12)\) \(\approx\) \(1.63385 + 0.462126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.838 + 0.544i)T \)
5 \( 1 + (-1.20 - 1.88i)T \)
7 \( 1 + (-2.05 - 1.66i)T \)
good3 \( 1 + (0.977 - 0.375i)T + (2.22 - 2.00i)T^{2} \)
11 \( 1 + (1.84 - 2.04i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.551 - 1.08i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.714 + 0.578i)T + (3.53 - 16.6i)T^{2} \)
19 \( 1 + (-7.58 + 3.37i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (1.25 + 1.92i)T + (-9.35 + 21.0i)T^{2} \)
29 \( 1 + (-3.05 - 4.21i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.82 - 0.191i)T + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (-0.812 - 0.0425i)T + (36.7 + 3.86i)T^{2} \)
41 \( 1 + (9.01 + 2.92i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (6.42 + 6.42i)T + 43iT^{2} \)
47 \( 1 + (-6.03 + 7.44i)T + (-9.77 - 45.9i)T^{2} \)
53 \( 1 + (2.66 + 6.94i)T + (-39.3 + 35.4i)T^{2} \)
59 \( 1 + (7.97 - 1.69i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (0.775 - 3.64i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (8.29 + 10.2i)T + (-13.9 + 65.5i)T^{2} \)
71 \( 1 + (4.41 - 3.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.0695 + 1.32i)T + (-72.6 + 7.63i)T^{2} \)
79 \( 1 + (-12.9 + 1.36i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (-7.98 + 1.26i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-11.8 - 2.52i)T + (81.3 + 36.1i)T^{2} \)
97 \( 1 + (0.462 + 0.0733i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.67257855546883153362931388862, −10.72601060293693889435110785400, −10.14050901169658046694890829638, −8.962059307969202727647364956110, −7.61531434176083738008541227949, −6.53365705046966841517607669746, −5.34630599991122869113561783560, −4.95948364407882365898346919696, −3.12074772197155876917097711441, −2.03940138300919589866179859113, 1.18540961411454916209193479067, 3.25183475564444479407992618151, 4.71861264076142025608663156279, 5.54424737530466616760519154527, 6.22318036137961966450923754857, 7.68167996999954852779165879070, 8.300705685788878181664841384785, 9.567570775956505302943992140724, 10.66040770240525801990344763551, 11.72208502628877389921688727272

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