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Properties

Label 2-306-17.2-c1-0-1
Degree $2$
Conductor $306$
Sign $-0.197 - 0.980i$
Analytic cond. $2.44342$
Root an. cond. $1.56314$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.585 + 1.41i)5-s + (−1.41 + 3.41i)7-s + (−0.707 + 0.707i)8-s + (−0.585 + 1.41i)10-s + (−0.292 − 0.121i)11-s − 4.82i·13-s + (−3.41 + 1.41i)14-s − 1.00·16-s + (−2.12 + 3.53i)17-s + (3.41 + 3.41i)19-s + (−1.41 + 0.585i)20-s + (−0.121 − 0.292i)22-s + (6.82 + 2.82i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.261 + 0.632i)5-s + (−0.534 + 1.29i)7-s + (−0.250 + 0.250i)8-s + (−0.185 + 0.447i)10-s + (−0.0883 − 0.0365i)11-s − 1.33i·13-s + (−0.912 + 0.377i)14-s − 0.250·16-s + (−0.514 + 0.857i)17-s + (0.783 + 0.783i)19-s + (−0.316 + 0.130i)20-s + (−0.0258 − 0.0624i)22-s + (1.42 + 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(306\)    =    \(2 \cdot 3^{2} \cdot 17\)
Sign: $-0.197 - 0.980i$
Analytic conductor: \(2.44342\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{306} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 306,\ (\ :1/2),\ -0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01264 + 1.23654i\)
\(L(\frac12)\) \(\approx\) \(1.01264 + 1.23654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
17 \( 1 + (2.12 - 3.53i)T \)
good5 \( 1 + (-0.585 - 1.41i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.41 - 3.41i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.292 + 0.121i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.82iT - 13T^{2} \)
19 \( 1 + (-3.41 - 3.41i)T + 19iT^{2} \)
23 \( 1 + (-6.82 - 2.82i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.41 + 3.41i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.41 + 0.585i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (4.82 - 2i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.36 + 5.70i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-7.24 + 7.24i)T - 43iT^{2} \)
47 \( 1 + 1.65iT - 47T^{2} \)
53 \( 1 + (3.41 + 3.41i)T + 53iT^{2} \)
59 \( 1 + (-2.41 + 2.41i)T - 59iT^{2} \)
61 \( 1 + (-1.17 + 2.82i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 + (-4.82 + 2i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.94 - 11.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (2 + 0.828i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (1.58 + 1.58i)T + 83iT^{2} \)
89 \( 1 - 0.928iT - 89T^{2} \)
97 \( 1 + (0.878 + 2.12i)T + (-68.5 + 68.5i)T^{2} \)
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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

−12.26901854362607560633738887287, −11.08768128174846436830855571925, −10.12152939155307970330818656851, −9.025571452455863763593113129636, −8.095584039147618887002413498679, −6.94552986867088722899185810186, −5.90772815613422143492255266102, −5.32481339551174074613179330486, −3.52120176524809938116631946451, −2.54799013953468199419492326413, 1.10053598697703898447853294427, 2.94490709165696495911500496698, 4.32145993712187888831128008693, 5.04332673863417242636374485872, 6.63264837609531143414042481273, 7.27292864928569052737342533383, 9.061378176061001012594324979059, 9.482962994765168560683906096910, 10.74335470414725335797637231260, 11.35923075950302700191642408601

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