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Properties

Label 2-308-11.3-c1-0-5
Degree $2$
Conductor $308$
Sign $-0.971 + 0.236i$
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  + (−0.585 − 1.80i)3-s + (−1.81 − 1.31i)5-s + (0.309 − 0.951i)7-s + (−0.472 + 0.343i)9-s + (−3.20 + 0.859i)11-s + (−3.19 + 2.32i)13-s + (−1.31 + 4.04i)15-s + (0.797 + 0.579i)17-s + (−1.91 − 5.89i)19-s − 1.89·21-s − 1.77·23-s + (0.0125 + 0.0387i)25-s + (−3.70 − 2.68i)27-s + (−2.39 + 7.36i)29-s + (7.41 − 5.39i)31-s + ⋯
L(s)  = 1  + (−0.337 − 1.03i)3-s + (−0.812 − 0.590i)5-s + (0.116 − 0.359i)7-s + (−0.157 + 0.114i)9-s + (−0.965 + 0.259i)11-s + (−0.885 + 0.643i)13-s + (−0.339 + 1.04i)15-s + (0.193 + 0.140i)17-s + (−0.439 − 1.35i)19-s − 0.413·21-s − 0.370·23-s + (0.00251 + 0.00774i)25-s + (−0.712 − 0.517i)27-s + (−0.444 + 1.36i)29-s + (1.33 − 0.968i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.971 + 0.236i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.971 + 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0760798 - 0.635090i\)
\(L(\frac12)\) \(\approx\) \(0.0760798 - 0.635090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (3.20 - 0.859i)T \)
good3 \( 1 + (0.585 + 1.80i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.81 + 1.31i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (3.19 - 2.32i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.797 - 0.579i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.91 + 5.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 + (2.39 - 7.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.41 + 5.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.976 + 3.00i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.554 + 1.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.78T + 43T^{2} \)
47 \( 1 + (3.48 + 10.7i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.624 + 0.454i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.80 + 5.54i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.31 - 3.13i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.97T + 67T^{2} \)
71 \( 1 + (-5.73 - 4.16i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.71 + 8.34i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.06 + 5.86i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.515 + 0.374i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.73T + 89T^{2} \)
97 \( 1 + (3.27 - 2.38i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.56157285508191307401290957901, −10.48969282278446418038108484100, −9.293333305198130476222403379436, −8.099803452250185759359875191206, −7.41430755560585078472811932308, −6.62511578180690669976256491668, −5.15108093649822269617592545781, −4.17322061587624831715597292046, −2.26841058119083248138248513812, −0.46969357494559519813488300294, 2.75439998969745688809241133926, 3.97485085632084365694936765082, 5.01883597273743757540822013524, 6.01855054747314362419681693410, 7.59808424629224380854935459353, 8.130814038975692532285660681028, 9.678753866130586286280244915753, 10.30418145978820588836821530194, 11.04125956722551622798361694930, 11.94567345609639766522254114063

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