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Properties

Label 2-308-77.62-c1-0-1
Degree $2$
Conductor $308$
Sign $0.400 - 0.916i$
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  + (−0.359 − 0.116i)3-s + (1.61 + 2.22i)5-s + (0.904 + 2.48i)7-s + (−2.31 − 1.67i)9-s + (−2.11 + 2.55i)11-s + (1.07 + 0.778i)13-s + (−0.321 − 0.989i)15-s + (2.01 − 1.46i)17-s + (−1.78 + 5.48i)19-s + (−0.0347 − 0.999i)21-s + 5.62·23-s + (−0.795 + 2.44i)25-s + (1.30 + 1.79i)27-s + (6.07 − 1.97i)29-s + (−0.830 + 1.14i)31-s + ⋯
L(s)  = 1  + (−0.207 − 0.0674i)3-s + (0.723 + 0.995i)5-s + (0.341 + 0.939i)7-s + (−0.770 − 0.559i)9-s + (−0.638 + 0.769i)11-s + (0.297 + 0.215i)13-s + (−0.0829 − 0.255i)15-s + (0.488 − 0.354i)17-s + (−0.409 + 1.25i)19-s + (−0.00758 − 0.218i)21-s + 1.17·23-s + (−0.159 + 0.489i)25-s + (0.250 + 0.344i)27-s + (1.12 − 0.366i)29-s + (−0.149 + 0.205i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.400 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07430 + 0.702479i\)
\(L(\frac12)\) \(\approx\) \(1.07430 + 0.702479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.904 - 2.48i)T \)
11 \( 1 + (2.11 - 2.55i)T \)
good3 \( 1 + (0.359 + 0.116i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.61 - 2.22i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.07 - 0.778i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.01 + 1.46i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.78 - 5.48i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.62T + 23T^{2} \)
29 \( 1 + (-6.07 + 1.97i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.830 - 1.14i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.224 - 0.690i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.87 + 5.78i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.58iT - 43T^{2} \)
47 \( 1 + (6.76 + 2.19i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.60 + 2.61i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (6.72 - 2.18i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.87 + 5.72i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.85T + 67T^{2} \)
71 \( 1 + (-6.22 + 4.52i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.51 - 4.67i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.03 + 6.93i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.86 + 7.16i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + (6.62 - 9.12i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.91319039558764072288035629355, −10.88629953026208793473996821798, −10.10125417427440328655731970534, −9.116238331509453795579038462648, −8.100258076265280961011100976970, −6.81249732776702675227047945059, −5.98634026434679871041306636150, −5.09174540226769492549504721056, −3.21609429145581746316007744316, −2.13222321754517821113748917503, 1.03347411044576443169492305539, 2.91812758209204116462006671122, 4.69124331116530929821270890220, 5.33903040386580139688862578656, 6.48391191360803653372318011950, 7.938006145950144096881164575549, 8.597066280691429125447755420248, 9.652168687835728542100056350112, 10.84714193366158875550837957925, 11.16707070250958305465498824383

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