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Properties

Label 2-306-17.8-c1-0-1
Degree $2$
Conductor $306$
Sign $0.815 - 0.578i$
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 + (3.41 + 1.41i)5-s + (1.41 − 0.585i)7-s + (0.707 + 0.707i)8-s + (−3.41 + 1.41i)10-s + (−1.70 − 4.12i)11-s − 0.828i·13-s + (−0.585 + 1.41i)14-s − 1.00·16-s + (2.12 + 3.53i)17-s + (0.585 − 0.585i)19-s + (1.41 − 3.41i)20-s + (4.12 + 1.70i)22-s + (1.17 + 2.82i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.52 + 0.632i)5-s + (0.534 − 0.221i)7-s + (0.250 + 0.250i)8-s + (−1.07 + 0.447i)10-s + (−0.514 − 1.24i)11-s − 0.229i·13-s + (−0.156 + 0.377i)14-s − 0.250·16-s + (0.514 + 0.857i)17-s + (0.134 − 0.134i)19-s + (0.316 − 0.763i)20-s + (0.878 + 0.363i)22-s + (0.244 + 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(306\)    =    \(2 \cdot 3^{2} \cdot 17\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(2.44342\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{306} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 306,\ (\ :1/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26680 + 0.403264i\)
\(L(\frac12)\) \(\approx\) \(1.26680 + 0.403264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 - 1.41i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.41 + 0.585i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.70 + 4.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.828iT - 13T^{2} \)
19 \( 1 + (-0.585 + 0.585i)T - 19iT^{2} \)
23 \( 1 + (-1.17 - 2.82i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.41 - 0.585i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.41 - 3.41i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.828 + 2i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (10.3 - 4.29i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.24 + 1.24i)T + 43iT^{2} \)
47 \( 1 + 9.65iT - 47T^{2} \)
53 \( 1 + (0.585 - 0.585i)T - 53iT^{2} \)
59 \( 1 + (0.414 + 0.414i)T + 59iT^{2} \)
61 \( 1 + (-6.82 + 2.82i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 7.41T + 67T^{2} \)
71 \( 1 + (0.828 - 2i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.94 + 2.05i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2 + 4.82i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (4.41 - 4.41i)T - 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + (5.12 + 2.12i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.45765490377870389126954947041, −10.53414703598304394194248582572, −10.07713529752803129713176649003, −8.917122250745625571876748997638, −8.058010689157198533342986684904, −6.86665930417664124773311000975, −5.90316040761339160580328428395, −5.23621820428728382401839697649, −3.15501054893724717482560229768, −1.60174187788080598049794235002, 1.55995254244719664547979385841, 2.57145032228850290058591757508, 4.64284374649415733665399655376, 5.44888302949348539239802536236, 6.82276436577544029777806803434, 8.009521617353248845076232002641, 9.069436744453029782466293544301, 9.774286071407184622221977629497, 10.37801267771894724014884951027, 11.65070742860131463025143899274

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