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Properties

Label 2-402-201.101-c1-0-7
Degree $2$
Conductor $402$
Sign $-0.595 - 0.803i$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
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  + (0.0475 + 0.998i)2-s + (1.01 + 1.40i)3-s + (−0.995 + 0.0950i)4-s + (1.12 + 1.30i)5-s + (−1.35 + 1.07i)6-s + (0.804 − 1.55i)7-s + (−0.142 − 0.989i)8-s + (−0.943 + 2.84i)9-s + (−1.24 + 1.19i)10-s + (−1.28 + 3.72i)11-s + (−1.14 − 1.30i)12-s + (0.482 − 0.613i)13-s + (1.59 + 0.728i)14-s + (−0.685 + 2.90i)15-s + (0.981 − 0.189i)16-s + (−0.240 + 2.52i)17-s + ⋯
L(s)  = 1  + (0.0336 + 0.706i)2-s + (0.585 + 0.810i)3-s + (−0.497 + 0.0475i)4-s + (0.505 + 0.583i)5-s + (−0.552 + 0.440i)6-s + (0.303 − 0.589i)7-s + (−0.0503 − 0.349i)8-s + (−0.314 + 0.949i)9-s + (−0.394 + 0.376i)10-s + (−0.388 + 1.12i)11-s + (−0.329 − 0.375i)12-s + (0.133 − 0.170i)13-s + (0.426 + 0.194i)14-s + (−0.176 + 0.750i)15-s + (0.245 − 0.0473i)16-s + (−0.0584 + 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $-0.595 - 0.803i$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{402} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ -0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760122 + 1.51023i\)
\(L(\frac12)\) \(\approx\) \(0.760122 + 1.51023i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0475 - 0.998i)T \)
3 \( 1 + (-1.01 - 1.40i)T \)
67 \( 1 + (8.16 - 0.562i)T \)
good5 \( 1 + (-1.12 - 1.30i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (-0.804 + 1.55i)T + (-4.06 - 5.70i)T^{2} \)
11 \( 1 + (1.28 - 3.72i)T + (-8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.482 + 0.613i)T + (-3.06 - 12.6i)T^{2} \)
17 \( 1 + (0.240 - 2.52i)T + (-16.6 - 3.21i)T^{2} \)
19 \( 1 + (-2.91 + 1.50i)T + (11.0 - 15.4i)T^{2} \)
23 \( 1 + (0.951 + 0.230i)T + (20.4 + 10.5i)T^{2} \)
29 \( 1 + (0.391 - 0.225i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.96 + 3.77i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (-1.32 + 2.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.890 + 1.25i)T + (-13.4 - 38.7i)T^{2} \)
43 \( 1 + (-0.325 + 0.148i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-1.08 + 1.13i)T + (-2.23 - 46.9i)T^{2} \)
53 \( 1 + (-2.13 + 4.68i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-12.0 + 1.73i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (-6.05 + 2.09i)T + (47.9 - 37.7i)T^{2} \)
71 \( 1 + (0.409 + 4.28i)T + (-69.7 + 13.4i)T^{2} \)
73 \( 1 + (-5.27 - 15.2i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (4.22 + 10.5i)T + (-57.1 + 54.5i)T^{2} \)
83 \( 1 + (1.22 + 6.35i)T + (-77.0 + 30.8i)T^{2} \)
89 \( 1 + (0.698 - 2.37i)T + (-74.8 - 48.1i)T^{2} \)
97 \( 1 + (-3.77 - 2.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.30262521941478731173507327745, −10.29610139965462654443983937897, −9.888230716470330069542924465042, −8.829996230931143925462197607703, −7.79223721218901342283710239932, −7.08667257924768581703434502415, −5.78576431025938595798238724021, −4.74012013682845683541822304280, −3.77863922764081151555316420108, −2.30526545133131716680771065052, 1.14733279365181342898853192951, 2.42095458740088088005468730034, 3.51296355771782502642340191760, 5.18688984999138811664372322522, 5.97569647756465992037985681126, 7.40872749212520121848090132289, 8.503006604429174797294688622769, 8.968486387007734140980359237134, 9.905213150025201677736795228034, 11.21061901073425265163976451691

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