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Properties

Label 2-308-7.2-c1-0-0
Degree $2$
Conductor $308$
Sign $-0.449 - 0.893i$
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  + (1.41 + 2.45i)3-s + (−0.599 + 1.03i)5-s + (−2.43 + 1.03i)7-s + (−2.51 + 4.35i)9-s + (−0.5 − 0.866i)11-s + 0.834·13-s − 3.39·15-s + (2.01 + 3.49i)17-s + (3.38 − 5.85i)19-s + (−5.99 − 4.50i)21-s + (0.0466 − 0.0808i)23-s + (1.78 + 3.08i)25-s − 5.76·27-s + 3.46·29-s + (−1.29 − 2.24i)31-s + ⋯
L(s)  = 1  + (0.818 + 1.41i)3-s + (−0.268 + 0.464i)5-s + (−0.919 + 0.392i)7-s + (−0.838 + 1.45i)9-s + (−0.150 − 0.261i)11-s + 0.231·13-s − 0.877·15-s + (0.489 + 0.847i)17-s + (0.775 − 1.34i)19-s + (−1.30 − 0.982i)21-s + (0.00973 − 0.0168i)23-s + (0.356 + 0.617i)25-s − 1.10·27-s + 0.644·29-s + (−0.233 − 0.403i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.449 - 0.893i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747364 + 1.21322i\)
\(L(\frac12)\) \(\approx\) \(0.747364 + 1.21322i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 - 1.03i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.41 - 2.45i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.599 - 1.03i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.834T + 13T^{2} \)
17 \( 1 + (-2.01 - 3.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.38 + 5.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0466 + 0.0808i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + (1.29 + 2.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.19 - 9.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.62T + 41T^{2} \)
43 \( 1 - 0.894T + 43T^{2} \)
47 \( 1 + (-0.453 + 0.785i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.58 - 2.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.31 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.03 + 8.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.23 + 7.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + (6.58 + 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.16 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.69T + 83T^{2} \)
89 \( 1 + (-7.29 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−11.80630397047861167950372965988, −10.77101599066898248323055055926, −10.09048013971122067597980093973, −9.212347351381367106321121473005, −8.566979632619752077763739280550, −7.29626672426834129931822403419, −5.96748810489535484839084126157, −4.72277885495451527730442654996, −3.46836250174504741851628794710, −2.88098589695771331205809345611, 1.02245028744057994114333633984, 2.65599432462213122010986453370, 3.82301785186045701289300936217, 5.62358873044832233680795221461, 6.83608079516567011878195520250, 7.50644271835377563121697762738, 8.371656452639561135849099648232, 9.322017281384519201358721763828, 10.31792424228220913903765901945, 11.85507017577444136421697685957

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