Properties

Degree $2$
Conductor $722$
Sign $-0.692 + 0.721i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.694 + 3.93i)5-s + (0.939 + 0.342i)6-s + (−1.5 − 2.59i)7-s + (0.500 − 0.866i)8-s + (−1.53 + 1.28i)9-s + (3.06 − 2.57i)10-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + (−0.939 − 0.342i)13-s + (−0.520 + 2.95i)14-s + (−0.694 − 3.93i)15-s + (−0.939 + 0.342i)16-s + (2.29 + 1.92i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (−0.310 + 1.76i)5-s + (0.383 + 0.139i)6-s + (−0.566 − 0.981i)7-s + (0.176 − 0.306i)8-s + (−0.510 + 0.428i)9-s + (0.968 − 0.813i)10-s + (−0.301 + 0.522i)11-s + (−0.144 − 0.250i)12-s + (−0.260 − 0.0948i)13-s + (−0.139 + 0.789i)14-s + (−0.179 − 1.01i)15-s + (−0.234 + 0.0855i)16-s + (0.557 + 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.692 + 0.721i$
Motivic weight: \(1\)
Character: $\chi_{722} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0255783 - 0.0599641i\)
\(L(\frac12)\) \(\approx\) \(0.0255783 - 0.0599641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (0.939 - 0.342i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (0.694 - 3.93i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.29 - 1.92i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.83 + 3.21i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (7.51 - 2.73i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.694 + 3.93i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.12 + 5.14i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (11.4 + 9.64i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.347 - 1.96i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.29 - 1.92i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.347 - 1.96i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (8.45 - 3.07i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (9.39 - 3.42i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.53 - 1.28i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.27220524978483852232087184085, −9.759652596163686693985900835750, −8.176919520529228865636645504063, −7.43865589774974124344822042769, −6.76707376370624722009347082490, −5.81919142140426388760559059716, −4.26829249353077784543512130388, −3.30805243101999857571337511118, −2.35459220916438817590796937151, −0.04662004402322542939171730073, 1.24288164526237256089342830280, 3.10573996021700404727701439850, 4.77918736764016040973335984865, 5.51038790679459080897304277447, 6.09383870462090113184152055314, 7.33941150893024973216001081455, 8.407848474687982473037327791890, 8.940665760035160260588928883283, 9.447460987651500883614459596638, 10.66746965965069453861414211771

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