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Properties

Label 2-308-7.2-c1-0-1
Degree $2$
Conductor $308$
Sign $-0.0996 - 0.995i$
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  + (0.794 + 1.37i)3-s + (−1.64 + 2.84i)5-s + (2.64 − 0.0963i)7-s + (0.238 − 0.413i)9-s + (0.5 + 0.866i)11-s − 4.98·13-s − 5.22·15-s + (1.84 + 3.20i)17-s + (−2.84 + 4.93i)19-s + (2.23 + 3.56i)21-s + (−0.349 + 0.605i)23-s + (−2.90 − 5.03i)25-s + 5.52·27-s + 7.68·29-s + (−5.25 − 9.10i)31-s + ⋯
L(s)  = 1  + (0.458 + 0.794i)3-s + (−0.735 + 1.27i)5-s + (0.999 − 0.0364i)7-s + (0.0795 − 0.137i)9-s + (0.150 + 0.261i)11-s − 1.38·13-s − 1.34·15-s + (0.448 + 0.777i)17-s + (−0.653 + 1.13i)19-s + (0.487 + 0.776i)21-s + (−0.0729 + 0.126i)23-s + (−0.581 − 1.00i)25-s + 1.06·27-s + 1.42·29-s + (−0.943 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0996 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0996 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.0996 - 0.995i$
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: \((2,\ 308,\ (\ :1/2),\ -0.0996 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936114 + 1.03453i\)
\(L(\frac12)\) \(\approx\) \(0.936114 + 1.03453i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.0963i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.794 - 1.37i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.64 - 2.84i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4.98T + 13T^{2} \)
17 \( 1 + (-1.84 - 3.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.84 - 4.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.349 - 0.605i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + (5.25 + 9.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.28 + 9.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.81T + 41T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 + (1.15 - 1.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.60 - 2.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.88 + 6.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.47 - 4.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.810 - 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.30T + 71T^{2} \)
73 \( 1 + (2.70 + 4.68i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.36 + 2.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−11.83854016422797573803060946047, −10.80673204150653615773195454113, −10.24171224551235887943590986490, −9.262791785899758622974151932251, −7.929928471919498375446508986556, −7.43301893057186764375547676831, −6.07477235983964078624930098792, −4.48123149648900932946715113191, −3.76968095752694427187523989997, −2.40265256290060261019015024870, 1.07083556439729716118656628580, 2.59786165038871904367700510929, 4.61233893173603829153451800146, 5.00566227729328721009000076616, 6.92401275326877987531542536111, 7.77785775276031277739374324995, 8.410501946090070817279948154678, 9.231844452979391084434663251360, 10.64052019105558136447448636790, 11.82611521227536084564425589218

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