Properties

Label 2-804-804.23-c1-0-88
Degree $2$
Conductor $804$
Sign $0.718 + 0.695i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (−1.39 − 0.210i)2-s + (1.49 + 0.878i)3-s + (1.91 + 0.587i)4-s + (1.84 − 0.844i)5-s + (−1.90 − 1.54i)6-s + (−2.22 − 2.32i)7-s + (−2.54 − 1.22i)8-s + (1.45 + 2.62i)9-s + (−2.76 + 0.792i)10-s + (−5.66 − 0.540i)11-s + (2.33 + 2.55i)12-s + (1.43 + 0.276i)13-s + (2.61 + 3.72i)14-s + (3.50 + 0.365i)15-s + (3.30 + 2.24i)16-s + (2.25 − 4.38i)17-s + ⋯
L(s)  = 1  + (−0.988 − 0.148i)2-s + (0.861 + 0.507i)3-s + (0.955 + 0.293i)4-s + (0.827 − 0.377i)5-s + (−0.776 − 0.629i)6-s + (−0.839 − 0.880i)7-s + (−0.901 − 0.432i)8-s + (0.484 + 0.874i)9-s + (−0.874 + 0.250i)10-s + (−1.70 − 0.163i)11-s + (0.674 + 0.738i)12-s + (0.397 + 0.0766i)13-s + (0.699 + 0.995i)14-s + (0.904 + 0.0942i)15-s + (0.827 + 0.561i)16-s + (0.547 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25417 - 0.507327i\)
\(L(\frac12)\) \(\approx\) \(1.25417 - 0.507327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.210i)T \)
3 \( 1 + (-1.49 - 0.878i)T \)
67 \( 1 + (8.01 - 1.65i)T \)
good5 \( 1 + (-1.84 + 0.844i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (2.22 + 2.32i)T + (-0.333 + 6.99i)T^{2} \)
11 \( 1 + (5.66 + 0.540i)T + (10.8 + 2.08i)T^{2} \)
13 \( 1 + (-1.43 - 0.276i)T + (12.0 + 4.83i)T^{2} \)
17 \( 1 + (-2.25 + 4.38i)T + (-9.86 - 13.8i)T^{2} \)
19 \( 1 + (-5.85 + 6.14i)T + (-0.904 - 18.9i)T^{2} \)
23 \( 1 + (-7.14 + 2.86i)T + (16.6 - 15.8i)T^{2} \)
29 \( 1 + (-0.448 - 0.258i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.08 + 5.63i)T + (-28.7 + 11.5i)T^{2} \)
37 \( 1 + (-3.45 - 5.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.17 + 0.437i)T + (40.8 - 3.89i)T^{2} \)
43 \( 1 + (0.737 + 1.14i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (-5.37 - 4.22i)T + (11.0 + 45.6i)T^{2} \)
53 \( 1 + (-0.952 + 1.48i)T + (-22.0 - 48.2i)T^{2} \)
59 \( 1 + (2.95 + 3.41i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (5.41 - 0.516i)T + (59.8 - 11.5i)T^{2} \)
71 \( 1 + (4.77 - 2.46i)T + (41.1 - 57.8i)T^{2} \)
73 \( 1 + (11.5 - 1.10i)T + (71.6 - 13.8i)T^{2} \)
79 \( 1 + (8.67 - 3.00i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (-3.55 + 4.98i)T + (-27.1 - 78.4i)T^{2} \)
89 \( 1 + (3.47 - 0.499i)T + (85.3 - 25.0i)T^{2} \)
97 \( 1 + (3.99 + 6.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.879212152783358264444001154413, −9.450989650576636422972917728411, −8.750043804701760853420891491661, −7.53006226487953148604219715270, −7.25121203754039411006647424912, −5.77845223552098858932197946248, −4.74571760153767887457463556059, −3.08121050845480936193671718365, −2.68700268841447042776304712658, −0.877739136303939548804685458120, 1.50698958051327042760714594834, 2.66149236802869221613419116501, 3.22273501520124083307827836830, 5.73344432951542624573930687965, 5.91473767624388743509312457039, 7.25289339206003601957914974586, 7.80539952246957089604554230191, 8.741190673018255486064196390108, 9.460191158128642226719616074064, 10.11783436476700108062284663449

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