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Properties

Label 2-348-348.347-c1-0-13
Degree $2$
Conductor $348$
Sign $-0.950 - 0.310i$
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  + (0.830 + 1.14i)2-s + 1.73i·3-s + (−0.621 + 1.90i)4-s + (−1.98 + 1.43i)6-s − 0.926i·7-s + (−2.69 + 0.866i)8-s − 2.99·9-s + 4.44i·11-s + (−3.29 − 1.07i)12-s − 2.72·13-s + (1.06 − 0.768i)14-s + (−3.22 − 2.36i)16-s + 8.24·17-s + (−2.49 − 3.43i)18-s + 1.60·21-s + (−5.08 + 3.68i)22-s + ⋯
L(s)  = 1  + (0.586 + 0.809i)2-s + 0.999i·3-s + (−0.310 + 0.950i)4-s + (−0.809 + 0.586i)6-s − 0.350i·7-s + (−0.951 + 0.306i)8-s − 0.999·9-s + 1.33i·11-s + (−0.950 − 0.310i)12-s − 0.755·13-s + (0.283 − 0.205i)14-s + (−0.806 − 0.590i)16-s + 1.99·17-s + (−0.586 − 0.809i)18-s + 0.350·21-s + (−1.08 + 0.786i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348\)    =    \(2^{2} \cdot 3 \cdot 29\)
Sign: $-0.950 - 0.310i$
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: \((2,\ 348,\ (\ :1/2),\ -0.950 - 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244310 + 1.53267i\)
\(L(\frac12)\) \(\approx\) \(0.244310 + 1.53267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.830 - 1.14i)T \)
3 \( 1 - 1.73iT \)
29 \( 1 + 5.38T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 + 0.926iT - 7T^{2} \)
11 \( 1 - 4.44iT - 11T^{2} \)
13 \( 1 + 2.72T + 13T^{2} \)
17 \( 1 - 8.24T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 13.6iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12.4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 5.03T + 89T^{2} \)
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

−12.20537586843100919733900345698, −10.95497110315064356469627348933, −9.841680567437775389753704464405, −9.280117894687901849490456713553, −7.891064573651600448977051861056, −7.25035892589771705182836343147, −5.84822016122030518957180121848, −4.94837709916123878567225696169, −4.11253501972986197760789381597, −2.90497477215471696216043422945, 0.949424901649530230871328579945, 2.54505249199239985655565783078, 3.52507451724828318401829042779, 5.36639112852967525559570142730, 5.86624062549234407970301513785, 7.18337981384980444214434549844, 8.326260519438936212512531709959, 9.280324082572086545014691131251, 10.42484538401881008740828506072, 11.37721128289864365700363923688

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