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Properties

Label 2-350-5.4-c5-0-4
Degree $2$
Conductor $350$
Sign $0.894 + 0.447i$
Analytic cond. $56.1343$
Root an. cond. $7.49228$
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  + 4i·2-s + 27.7i·3-s − 16·4-s − 111.·6-s + 49i·7-s − 64i·8-s − 528.·9-s − 481.·11-s − 444. i·12-s + 883. i·13-s − 196·14-s + 256·16-s + 1.84e3i·17-s − 2.11e3i·18-s − 2.44e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.78i·3-s − 0.5·4-s − 1.25·6-s + 0.377i·7-s − 0.353i·8-s − 2.17·9-s − 1.20·11-s − 0.890i·12-s + 1.44i·13-s − 0.267·14-s + 0.250·16-s + 1.54i·17-s − 1.53i·18-s − 1.55·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(56.1343\)
Root analytic conductor: \(7.49228\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6632653203\)
\(L(\frac12)\) \(\approx\) \(0.6632653203\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 \)
7 \( 1 - 49iT \)
good3 \( 1 - 27.7iT - 243T^{2} \)
11 \( 1 + 481.T + 1.61e5T^{2} \)
13 \( 1 - 883. iT - 3.71e5T^{2} \)
17 \( 1 - 1.84e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.44e3T + 2.47e6T^{2} \)
23 \( 1 - 169. iT - 6.43e6T^{2} \)
29 \( 1 - 554.T + 2.05e7T^{2} \)
31 \( 1 - 7.73e3T + 2.86e7T^{2} \)
37 \( 1 + 1.14e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.08e4T + 1.15e8T^{2} \)
43 \( 1 - 1.53e4iT - 1.47e8T^{2} \)
47 \( 1 + 9.70e3iT - 2.29e8T^{2} \)
53 \( 1 - 8.82e3iT - 4.18e8T^{2} \)
59 \( 1 - 4.05e4T + 7.14e8T^{2} \)
61 \( 1 + 1.67e3T + 8.44e8T^{2} \)
67 \( 1 + 9.88e3iT - 1.35e9T^{2} \)
71 \( 1 + 5.13e4T + 1.80e9T^{2} \)
73 \( 1 - 2.67e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.02e4T + 3.07e9T^{2} \)
83 \( 1 - 5.86e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.83e4T + 5.58e9T^{2} \)
97 \( 1 - 1.17e5iT - 8.58e9T^{2} \)
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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

−11.09606264793770821663096358523, −10.47710521883925905892511298442, −9.620139949941686559932901420999, −8.745069695878748984142933494149, −8.124784680913099187445804614100, −6.45437031687514284148176959481, −5.60528480254235918756723448389, −4.52251976129638503430415326765, −3.96655020653287329841279817025, −2.42349055991471170394777886656, 0.21188292401693687533014573619, 0.876815593105370713034360240402, 2.34373594272948748649180122731, 2.95958831236683484805822165531, 4.87332042205294855271865962405, 5.96182220342590275281118041601, 7.08572635261470908529438223223, 7.942179604698106485342864326381, 8.531557050511597136631270119396, 10.06677211504099269313115311718

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