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Properties

Label 2-348-29.25-c1-0-3
Degree $2$
Conductor $348$
Sign $-0.449 + 0.893i$
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  + (−0.900 + 0.433i)3-s + (0.655 − 2.86i)5-s + (−2.50 + 1.20i)7-s + (0.623 − 0.781i)9-s + (−1.21 − 1.52i)11-s + (−3.57 − 4.47i)13-s + (0.655 + 2.86i)15-s + 2.36·17-s + (−3.32 − 1.59i)19-s + (1.73 − 2.16i)21-s + (0.153 + 0.674i)23-s + (−3.30 − 1.59i)25-s + (−0.222 + 0.974i)27-s + (−4.13 − 3.44i)29-s + (1.58 − 6.94i)31-s + ⋯
L(s)  = 1  + (−0.520 + 0.250i)3-s + (0.292 − 1.28i)5-s + (−0.945 + 0.455i)7-s + (0.207 − 0.260i)9-s + (−0.367 − 0.460i)11-s + (−0.990 − 1.24i)13-s + (0.169 + 0.740i)15-s + 0.574·17-s + (−0.762 − 0.366i)19-s + (0.377 − 0.473i)21-s + (0.0321 + 0.140i)23-s + (−0.660 − 0.318i)25-s + (−0.0428 + 0.187i)27-s + (−0.768 − 0.639i)29-s + (0.284 − 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348\)    =    \(2^{2} \cdot 3 \cdot 29\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(2.77879\)
Root analytic conductor: \(1.66697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{348} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 348,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369747 - 0.599969i\)
\(L(\frac12)\) \(\approx\) \(0.369747 - 0.599969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (4.13 + 3.44i)T \)
good5 \( 1 + (-0.655 + 2.86i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (2.50 - 1.20i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (1.21 + 1.52i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.57 + 4.47i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 + (3.32 + 1.59i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.153 - 0.674i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-1.58 + 6.94i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-2.45 + 3.07i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 3.55T + 41T^{2} \)
43 \( 1 + (1.19 + 5.25i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-4.03 - 5.06i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.78 - 7.83i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + (1.09 - 0.529i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (9.03 - 11.3i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-0.992 - 1.24i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-3.52 - 15.4i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (0.407 - 0.510i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-9.22 - 4.44i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-2.94 + 12.8i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-1.74 - 0.842i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.22624452478852004480529972317, −10.07213208985983645994284982276, −9.495652643601816315093347480647, −8.533446596888313878582011558022, −7.45610260778524345963361493144, −5.88072248628996946253790146134, −5.49810053674996572519801079966, −4.26664006620998135882071087696, −2.68698892500010233582498328722, −0.49534562868121193793929867629, 2.18709681849934172657297268242, 3.49089321557909741890163700195, 4.91646656662252890003407259309, 6.38156276438218991904809247944, 6.80630471972165048227214225529, 7.65932308042583415867938237445, 9.320412259397426351614015591497, 10.17184379654636590584468930758, 10.68390562765075040801091264888, 11.80533294373778647345973986652

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