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Properties

Label 2-348-348.347-c1-0-14
Degree $2$
Conductor $348$
Sign $-0.962 + 0.270i$
Analytic cond. $2.77879$
Root an. cond. $1.66697$
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  + 1.41i·2-s + (0.468 + 1.66i)3-s − 2.00·4-s + 4.45i·5-s + (−2.35 + 0.663i)6-s − 2.82i·8-s + (−2.56 + 1.56i)9-s − 6.29·10-s − 3.90i·11-s + (−0.937 − 3.33i)12-s + 4.31·13-s + (−7.42 + 2.08i)15-s + 4.00·16-s + (−2.21 − 3.62i)18-s + 7.61·19-s − 8.90i·20-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.270 + 0.962i)3-s − 1.00·4-s + 1.99i·5-s + (−0.962 + 0.270i)6-s − 1.00i·8-s + (−0.853 + 0.521i)9-s − 1.99·10-s − 1.17i·11-s + (−0.270 − 0.962i)12-s + 1.19·13-s + (−1.91 + 0.539i)15-s + 1.00·16-s + (−0.521 − 0.853i)18-s + 1.74·19-s − 1.99i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348\)    =    \(2^{2} \cdot 3 \cdot 29\)
Sign: $-0.962 + 0.270i$
Analytic conductor: \(2.77879\)
Root analytic conductor: \(1.66697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{348} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 348,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172488 - 1.25042i\)
\(L(\frac12)\) \(\approx\) \(0.172488 - 1.25042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-0.468 - 1.66i)T \)
29 \( 1 - 5.38iT \)
good5 \( 1 - 4.45iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 3.90iT - 11T^{2} \)
13 \( 1 - 4.31T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.61T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
31 \( 1 + 3.48T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 9.11T + 43T^{2} \)
47 \( 1 + 2.19iT - 47T^{2} \)
53 \( 1 - 0.474iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.58500619341951581727725169397, −10.86878675287511113286386247528, −10.14845401819687123947329240292, −9.166799807904119497270978363797, −8.179774104053151163599583849879, −7.20593581760173157316900337346, −6.18657000756367799065363270062, −5.41568230779287328241506294869, −3.61602216387184255157513279036, −3.24432506719838213197829996595, 0.949438472866915211484016234343, 1.88521682338592601522507627047, 3.68710274946788999886720858811, 4.88257372856060550693713237717, 5.79165719049979829376095732326, 7.58713988792562171158175920510, 8.383586163771274376880456968602, 9.163715096449820835362403331424, 9.813016759901623193364487557787, 11.45086863992764398526656356807

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