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Properties

Label 2-294-7.2-c5-0-18
Degree $2$
Conductor $294$
Sign $0.947 + 0.318i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (−30.5 + 52.8i)5-s − 36·6-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (−122. − 211. i)10-s + (18.2 + 31.6i)11-s + (72 − 124. i)12-s − 34.5·13-s − 549.·15-s + (−128 + 221. i)16-s + (−1.03e3 − 1.78e3i)17-s + (−162 − 280. i)18-s + (226. − 391. i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.546 + 0.946i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.386 − 0.669i)10-s + (0.0455 + 0.0788i)11-s + (0.144 − 0.249i)12-s − 0.0567·13-s − 0.630·15-s + (−0.125 + 0.216i)16-s + (−0.864 − 1.49i)17-s + (−0.117 − 0.204i)18-s + (0.143 − 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7498207336\)
\(L(\frac12)\) \(\approx\) \(0.7498207336\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 3.46i)T \)
3 \( 1 + (-4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (30.5 - 52.8i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-18.2 - 31.6i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 34.5T + 3.71e5T^{2} \)
17 \( 1 + (1.03e3 + 1.78e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-226. + 391. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (842. - 1.45e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 4.76e3T + 2.05e7T^{2} \)
31 \( 1 + (2.63e3 + 4.55e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-6.41e3 + 1.11e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 7.12e3T + 1.15e8T^{2} \)
43 \( 1 - 1.11e4T + 1.47e8T^{2} \)
47 \( 1 + (1.17e4 - 2.03e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-3.51e3 - 6.08e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.21e4 + 3.82e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-9.69e3 + 1.67e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.04e4 + 1.81e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.98e4T + 1.80e9T^{2} \)
73 \( 1 + (1.85e4 + 3.20e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.10e4 - 3.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 6.31e3T + 3.93e9T^{2} \)
89 \( 1 + (2.57e4 - 4.45e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.27e5T + 8.58e9T^{2} \)
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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

−11.10013684943588723215929444153, −9.618036268444725818201688636104, −9.182925359377721862656260444208, −7.72862656859814840586007141855, −7.26832845584367800222180109021, −6.06255575862572756083903982230, −4.79588065535016770624800005026, −3.63970699014396584381104725703, −2.37213439224294875715512849939, −0.25784404514436090916783576927, 1.00948000079525689390794197276, 2.11931261879410886376081098485, 3.63877585010834601417724967217, 4.59906983269689322257651654206, 6.08281657980994776768704224692, 7.40189167177027544402155400617, 8.429926355187836498322276458573, 8.800595384431472337308056469854, 10.04069961796474880417794398804, 11.07567510895942704842644818402

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