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Properties

Label 2-308-11.2-c2-0-9
Degree $2$
Conductor $308$
Sign $0.309 + 0.950i$
Analytic cond. $8.39239$
Root an. cond. $2.89696$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.11 − 2.98i)3-s + (0.466 + 1.43i)5-s + (1.55 − 2.14i)7-s + (5.20 − 16.0i)9-s + (−3.63 − 10.3i)11-s + (−2.30 − 0.749i)13-s + (6.20 + 4.51i)15-s + (1.91 − 0.622i)17-s + (1.13 + 1.55i)19-s − 13.4i·21-s + 40.7·23-s + (18.3 − 13.3i)25-s + (−12.3 − 37.9i)27-s + (−25.2 + 34.7i)29-s + (−11.1 + 34.3i)31-s + ⋯
L(s)  = 1  + (1.37 − 0.996i)3-s + (0.0932 + 0.287i)5-s + (0.222 − 0.305i)7-s + (0.578 − 1.78i)9-s + (−0.330 − 0.943i)11-s + (−0.177 − 0.0576i)13-s + (0.413 + 0.300i)15-s + (0.112 − 0.0366i)17-s + (0.0596 + 0.0820i)19-s − 0.640i·21-s + 1.77·23-s + (0.735 − 0.534i)25-s + (−0.457 − 1.40i)27-s + (−0.871 + 1.19i)29-s + (−0.360 + 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(8.39239\)
Root analytic conductor: \(2.89696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.09641 - 1.52235i\)
\(L(\frac12)\) \(\approx\) \(2.09641 - 1.52235i\)
\(L(2)\) 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 \( 1 + (-1.55 + 2.14i)T \)
11 \( 1 + (3.63 + 10.3i)T \)
good3 \( 1 + (-4.11 + 2.98i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (-0.466 - 1.43i)T + (-20.2 + 14.6i)T^{2} \)
13 \( 1 + (2.30 + 0.749i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-1.91 + 0.622i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-1.13 - 1.55i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 40.7T + 529T^{2} \)
29 \( 1 + (25.2 - 34.7i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (11.1 - 34.3i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (38.4 + 27.9i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (10.0 + 13.8i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 16.5iT - 1.84e3T^{2} \)
47 \( 1 + (-17.7 + 12.8i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (14.4 - 44.5i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-41.1 - 29.8i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-63.7 + 20.7i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 13.9T + 4.48e3T^{2} \)
71 \( 1 + (-33.0 - 101. i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (49.6 - 68.4i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (13.8 + 4.50i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (146. - 47.6i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 28.9T + 7.92e3T^{2} \)
97 \( 1 + (-35.1 + 108. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.26210698948491867124396611102, −10.36735855788223023060912361958, −8.939518943681693845114748306502, −8.569082557816369503110469366187, −7.34971055004328351438776271741, −6.87798665712870252325078474277, −5.32610768675822734881636739484, −3.53002427594080891482955527012, −2.69235489042724690633438982852, −1.21408753986278772534062628691, 2.07355222943248926890586258297, 3.22905417683721456453119769575, 4.46203012020215999063776469771, 5.27429927733944202613389455328, 7.12815838155170944154720397374, 8.075695539779586845178818203590, 9.017042282007225517833527172551, 9.572477070679602146940157451267, 10.44865135652962386169987220352, 11.53256662891035838762555594726

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