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Properties

Label 2-2646-63.20-c1-0-36
Degree $2$
Conductor $2646$
Sign $0.361 + 0.932i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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 + (0.499 + 0.866i)4-s + (−0.0338 − 0.0585i)5-s + 0.999i·8-s − 0.0676i·10-s + (−3.40 − 1.96i)11-s + (−3.32 + 1.92i)13-s + (−0.5 + 0.866i)16-s − 1.55·17-s − 5.84i·19-s + (0.0338 − 0.0585i)20-s + (−1.96 − 3.40i)22-s + (4.78 − 2.76i)23-s + (2.49 − 4.32i)25-s − 3.84·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0151 − 0.0261i)5-s + 0.353i·8-s − 0.0213i·10-s + (−1.02 − 0.592i)11-s + (−0.922 + 0.532i)13-s + (−0.125 + 0.216i)16-s − 0.376·17-s − 1.34i·19-s + (0.00755 − 0.0130i)20-s + (−0.418 − 0.725i)22-s + (0.998 − 0.576i)23-s + (0.499 − 0.865i)25-s − 0.753·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.567233627\)
\(L(\frac12)\) \(\approx\) \(1.567233627\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0338 + 0.0585i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.40 + 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.32 - 1.92i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
23 \( 1 + (-4.78 + 2.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.20 + 0.697i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.09 - 0.632i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 + (5.17 + 8.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.735 + 1.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.77 + 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.26iT - 53T^{2} \)
59 \( 1 + (4.70 + 8.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0705 + 0.0407i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.67 - 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.30iT - 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.93 + 6.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (-0.363 - 0.209i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.638710072002559370529334408933, −7.86619546769593215030903483495, −6.99380275500884133478444721667, −6.56217573726842802045976490429, −5.38239519679125038355722893680, −4.92705903529769083727535614707, −4.08669125344331550673458055583, −2.86604596693391429938748806975, −2.32865649046082447527791471147, −0.40101147780726385027765374811, 1.36633101954578201314450125180, 2.53174174395993565499253326019, 3.20465055281241146798540729827, 4.32877980724368327174719789104, 5.08560596835853244890512616536, 5.66140651207471031529331166359, 6.66584246638542392450129246216, 7.56776151504510394176827156692, 7.972963980226084504703143575180, 9.262417342162104103412353859322

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