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Properties

Label 2-308-77.60-c1-0-7
Degree $2$
Conductor $308$
Sign $0.691 + 0.722i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (2.27 − 1.01i)3-s + (−1.34 + 0.285i)5-s + (1.38 − 2.25i)7-s + (2.14 − 2.37i)9-s + (−0.893 − 3.19i)11-s + (0.390 + 1.20i)13-s + (−2.77 + 2.01i)15-s + (4.95 + 5.50i)17-s + (0.562 − 5.35i)19-s + (0.876 − 6.52i)21-s + (3.03 + 5.25i)23-s + (−2.84 + 1.26i)25-s + (0.153 − 0.473i)27-s + (−5.17 + 3.75i)29-s + (−8.06 − 1.71i)31-s + ⋯
L(s)  = 1  + (1.31 − 0.584i)3-s + (−0.601 + 0.127i)5-s + (0.524 − 0.851i)7-s + (0.713 − 0.792i)9-s + (−0.269 − 0.963i)11-s + (0.108 + 0.333i)13-s + (−0.715 + 0.519i)15-s + (1.20 + 1.33i)17-s + (0.129 − 1.22i)19-s + (0.191 − 1.42i)21-s + (0.633 + 1.09i)23-s + (−0.568 + 0.252i)25-s + (0.0296 − 0.0911i)27-s + (−0.960 + 0.697i)29-s + (−1.44 − 0.307i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69119 - 0.722601i\)
\(L(\frac12)\) \(\approx\) \(1.69119 - 0.722601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.38 + 2.25i)T \)
11 \( 1 + (0.893 + 3.19i)T \)
good3 \( 1 + (-2.27 + 1.01i)T + (2.00 - 2.22i)T^{2} \)
5 \( 1 + (1.34 - 0.285i)T + (4.56 - 2.03i)T^{2} \)
13 \( 1 + (-0.390 - 1.20i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.95 - 5.50i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (-0.562 + 5.35i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-3.03 - 5.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.17 - 3.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (8.06 + 1.71i)T + (28.3 + 12.6i)T^{2} \)
37 \( 1 + (-1.63 - 0.726i)T + (24.7 + 27.4i)T^{2} \)
41 \( 1 + (-1.43 - 1.04i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (0.357 - 3.40i)T + (-45.9 - 9.77i)T^{2} \)
53 \( 1 + (4.74 + 1.00i)T + (48.4 + 21.5i)T^{2} \)
59 \( 1 + (0.400 + 3.81i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + (2.51 - 0.535i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (5.64 - 9.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.44 - 13.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.168 - 1.60i)T + (-71.4 + 15.1i)T^{2} \)
79 \( 1 + (-3.17 + 3.52i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-2.36 + 7.26i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.77 + 3.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.65 + 14.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.39286208046946942514592416165, −10.85761695115585580995283066076, −9.432456760349836587470539414632, −8.566263896543259453401379361352, −7.64017158178476980068171091255, −7.30915041699158106618963083173, −5.63991567922704308447296451125, −3.94311530694837622070732356241, −3.18237784192809340337088550950, −1.48128060993082135743883794234, 2.22234913447030950571888938687, 3.38233608666761276234685999969, 4.51281168139217090532406526530, 5.63376900508352586931137347817, 7.59224500151656600220725588305, 7.924732555328403112283083724906, 9.087711453107840999841551624832, 9.641356040396838301135664507148, 10.75155936123944178829120806687, 12.01180548473229510702168523649

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