Properties

Label 2-354-59.7-c1-0-6
Degree $2$
Conductor $354$
Sign $0.587 + 0.808i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  + (−0.561 − 0.827i)2-s + (−0.0541 + 0.998i)3-s + (−0.370 + 0.928i)4-s + (3.87 − 1.79i)5-s + (0.856 − 0.515i)6-s + (0.929 − 3.34i)7-s + (0.976 − 0.214i)8-s + (−0.994 − 0.108i)9-s + (−3.66 − 2.20i)10-s + (−3.98 + 3.77i)11-s + (−0.907 − 0.419i)12-s + (3.32 − 0.361i)13-s + (−3.29 + 1.10i)14-s + (1.58 + 3.96i)15-s + (−0.725 − 0.687i)16-s + (−1.01 − 3.64i)17-s + ⋯
L(s)  = 1  + (−0.396 − 0.585i)2-s + (−0.0312 + 0.576i)3-s + (−0.185 + 0.464i)4-s + (1.73 − 0.802i)5-s + (0.349 − 0.210i)6-s + (0.351 − 1.26i)7-s + (0.345 − 0.0760i)8-s + (−0.331 − 0.0360i)9-s + (−1.15 − 0.696i)10-s + (−1.20 + 1.13i)11-s + (−0.261 − 0.121i)12-s + (0.922 − 0.100i)13-s + (−0.880 + 0.296i)14-s + (0.408 + 1.02i)15-s + (−0.181 − 0.171i)16-s + (−0.245 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.587 + 0.808i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.587 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23766 - 0.630458i\)
\(L(\frac12)\) \(\approx\) \(1.23766 - 0.630458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.561 + 0.827i)T \)
3 \( 1 + (0.0541 - 0.998i)T \)
59 \( 1 + (7.65 - 0.649i)T \)
good5 \( 1 + (-3.87 + 1.79i)T + (3.23 - 3.81i)T^{2} \)
7 \( 1 + (-0.929 + 3.34i)T + (-5.99 - 3.60i)T^{2} \)
11 \( 1 + (3.98 - 3.77i)T + (0.595 - 10.9i)T^{2} \)
13 \( 1 + (-3.32 + 0.361i)T + (12.6 - 2.79i)T^{2} \)
17 \( 1 + (1.01 + 3.64i)T + (-14.5 + 8.76i)T^{2} \)
19 \( 1 + (1.76 - 1.33i)T + (5.08 - 18.3i)T^{2} \)
23 \( 1 + (-2.05 + 3.87i)T + (-12.9 - 19.0i)T^{2} \)
29 \( 1 + (1.97 - 2.91i)T + (-10.7 - 26.9i)T^{2} \)
31 \( 1 + (-5.92 - 4.50i)T + (8.29 + 29.8i)T^{2} \)
37 \( 1 + (-0.148 - 0.0326i)T + (33.5 + 15.5i)T^{2} \)
41 \( 1 + (-0.987 - 1.86i)T + (-23.0 + 33.9i)T^{2} \)
43 \( 1 + (-3.59 - 3.40i)T + (2.32 + 42.9i)T^{2} \)
47 \( 1 + (1.05 + 0.489i)T + (30.4 + 35.8i)T^{2} \)
53 \( 1 + (9.25 - 5.56i)T + (24.8 - 46.8i)T^{2} \)
61 \( 1 + (-3.52 - 5.20i)T + (-22.5 + 56.6i)T^{2} \)
67 \( 1 + (13.3 - 2.94i)T + (60.8 - 28.1i)T^{2} \)
71 \( 1 + (-0.919 - 0.425i)T + (45.9 + 54.1i)T^{2} \)
73 \( 1 + (-13.5 + 4.57i)T + (58.1 - 44.1i)T^{2} \)
79 \( 1 + (0.470 + 8.67i)T + (-78.5 + 8.54i)T^{2} \)
83 \( 1 + (0.183 + 1.12i)T + (-78.6 + 26.5i)T^{2} \)
89 \( 1 + (5.68 - 8.38i)T + (-32.9 - 82.6i)T^{2} \)
97 \( 1 + (-2.26 - 0.763i)T + (77.2 + 58.7i)T^{2} \)
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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

−10.70861290389683099211726108420, −10.51080776219311574434488530543, −9.654611653140055335794708054187, −8.891872018908432927110271197836, −7.79536864640737665278546065698, −6.46607824985847296792075865708, −5.06832014016670320855707363546, −4.47699140640146726378951278961, −2.65200354163747462035921492434, −1.28278222077978663258138370065, 1.83414159339156877465574317114, 2.84101597713962738052171242570, 5.41527114296541630110729192527, 5.91195950692403021234186150862, 6.51949056183665606764606544817, 7.978323703143598887512176398278, 8.733784512502920981531779459615, 9.594584627901206110451478844601, 10.73488600686823923582323763598, 11.23855898021016889643383287347

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