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Properties

Label 2-351-117.49-c1-0-2
Degree $2$
Conductor $351$
Sign $0.930 - 0.366i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (−0.677 − 0.391i)2-s + (−0.693 − 1.20i)4-s + (0.0536 + 0.0309i)5-s + 3.75i·7-s + 2.65i·8-s + (−0.0242 − 0.0419i)10-s + (1.11 + 0.641i)11-s + (1.65 + 3.20i)13-s + (1.46 − 2.54i)14-s + (−0.349 + 0.606i)16-s + (2.74 − 4.75i)17-s + (2.72 + 1.57i)19-s − 0.0858i·20-s + (−0.501 − 0.869i)22-s + 5.65·23-s + ⋯
L(s)  = 1  + (−0.479 − 0.276i)2-s + (−0.346 − 0.600i)4-s + (0.0239 + 0.0138i)5-s + 1.41i·7-s + 0.937i·8-s + (−0.00766 − 0.0132i)10-s + (0.334 + 0.193i)11-s + (0.458 + 0.888i)13-s + (0.392 − 0.680i)14-s + (−0.0874 + 0.151i)16-s + (0.665 − 1.15i)17-s + (0.624 + 0.360i)19-s − 0.0192i·20-s + (−0.107 − 0.185i)22-s + 1.18·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.930 - 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936200 + 0.177772i\)
\(L(\frac12)\) \(\approx\) \(0.936200 + 0.177772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-1.65 - 3.20i)T \)
good2 \( 1 + (0.677 + 0.391i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0536 - 0.0309i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.75iT - 7T^{2} \)
11 \( 1 + (-1.11 - 0.641i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.74 + 4.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.72 - 1.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + (3.56 - 6.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.62 - 2.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.03 - 2.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.88iT - 41T^{2} \)
43 \( 1 - 4.22T + 43T^{2} \)
47 \( 1 + (2.73 - 1.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.752T + 53T^{2} \)
59 \( 1 + (-0.310 + 0.179i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + 1.65iT - 67T^{2} \)
71 \( 1 + (11.3 + 6.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.09iT - 73T^{2} \)
79 \( 1 + (0.616 + 1.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.0 + 6.38i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.30 - 0.751i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−11.64572829205850025093987080429, −10.54092372577460991463224590034, −9.339096174172376142794253807624, −9.185239721508571658623066367441, −8.077899744848838472811303637576, −6.64830470079874917914392853674, −5.59258111579793704299731331828, −4.76857533540046348871719318362, −2.93185346012425682775844136205, −1.51522759713645273380786146413, 0.901055593219564176107411746719, 3.40753038891319989367766819521, 4.10022845204895586219268755150, 5.64549485287456039472866357834, 7.01312111337075498697404301287, 7.65049979848629289910097191513, 8.503210380938900657353852639536, 9.559225130392570946399072089314, 10.38721565926982178958631516306, 11.25275652317113369193406489035

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