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Properties

Label 2-294-49.37-c1-0-5
Degree $2$
Conductor $294$
Sign $0.500 + 0.865i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
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  + (0.826 − 0.563i)2-s + (0.733 − 0.680i)3-s + (0.365 − 0.930i)4-s + (−0.409 − 0.126i)5-s + (0.222 − 0.974i)6-s + (2.34 + 1.22i)7-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (−0.409 + 0.126i)10-s + (−0.206 − 2.76i)11-s + (−0.365 − 0.930i)12-s + (0.793 + 0.381i)13-s + (2.62 − 0.311i)14-s + (−0.386 + 0.185i)15-s + (−0.733 − 0.680i)16-s + (−0.0792 − 0.0119i)17-s + ⋯
L(s)  = 1  + (0.584 − 0.398i)2-s + (0.423 − 0.392i)3-s + (0.182 − 0.465i)4-s + (−0.183 − 0.0564i)5-s + (0.0908 − 0.398i)6-s + (0.886 + 0.462i)7-s + (−0.0786 − 0.344i)8-s + (0.0249 − 0.332i)9-s + (−0.129 + 0.0399i)10-s + (−0.0624 − 0.832i)11-s + (−0.105 − 0.268i)12-s + (0.219 + 0.105i)13-s + (0.702 − 0.0831i)14-s + (−0.0996 + 0.0480i)15-s + (−0.183 − 0.170i)16-s + (−0.0192 − 0.00289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76455 - 1.01804i\)
\(L(\frac12)\) \(\approx\) \(1.76455 - 1.01804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
3 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-2.34 - 1.22i)T \)
good5 \( 1 + (0.409 + 0.126i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.206 + 2.76i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (-0.793 - 0.381i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (0.0792 + 0.0119i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (1.44 - 2.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.91 - 0.289i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (2.03 - 2.54i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-3.38 - 5.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.46 - 6.28i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.04 - 4.58i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.279 - 1.22i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (3.57 - 2.43i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-4.13 + 10.5i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (2.96 - 0.913i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-1.09 - 2.78i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (4.61 + 7.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.93 + 7.44i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-9.56 - 6.52i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (5.42 - 9.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.56 - 4.60i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.366 - 4.89i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 5.15T + 97T^{2} \)
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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

−11.72101178800006555950849818318, −10.97970655277595391838646570600, −9.832396399738097093523130954171, −8.552263052080907733428771816503, −7.988573630072959950465433941640, −6.53948805082530698875138520180, −5.52788147521661419668893460084, −4.28830335510772573718872364189, −3.01183664757381850340427769229, −1.60929110905398588708429035495, 2.22629697696733505465697879011, 3.89792703588349561399471359795, 4.61290712244001365970527883511, 5.82163014624651444841415114460, 7.27011708420843537019790234908, 7.86626628217003544925079848530, 8.968620824777856550342936166701, 10.10370448639439466949155411624, 11.10449246079193142634628096408, 11.91177563200888339885324018734

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