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Properties

Label 2-350-5.4-c5-0-26
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 − 7.77i·3-s − 16·4-s + 31.1·6-s + 49i·7-s − 64i·8-s + 182.·9-s − 588.·11-s + 124. i·12-s − 147. i·13-s − 196·14-s + 256·16-s + 63.1i·17-s + 730. i·18-s + 1.61e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.498i·3-s − 0.5·4-s + 0.352·6-s + 0.377i·7-s − 0.353i·8-s + 0.751·9-s − 1.46·11-s + 0.249i·12-s − 0.242i·13-s − 0.267·14-s + 0.250·16-s + 0.0530i·17-s + 0.531i·18-s + 1.02·19-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

\(L(3)\) \(\approx\) \(1.569110347\)
\(L(\frac12)\) \(\approx\) \(1.569110347\)
\(L(\frac{7}{2})\) 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 \( 1 + 7.77iT - 243T^{2} \)
11 \( 1 + 588.T + 1.61e5T^{2} \)
13 \( 1 + 147. iT - 3.71e5T^{2} \)
17 \( 1 - 63.1iT - 1.41e6T^{2} \)
19 \( 1 - 1.61e3T + 2.47e6T^{2} \)
23 \( 1 - 1.48e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.69e3T + 2.05e7T^{2} \)
31 \( 1 + 7.44e3T + 2.86e7T^{2} \)
37 \( 1 - 2.43e3iT - 6.93e7T^{2} \)
41 \( 1 - 334.T + 1.15e8T^{2} \)
43 \( 1 + 1.19e4iT - 1.47e8T^{2} \)
47 \( 1 + 5.86e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.50e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.23e4T + 7.14e8T^{2} \)
61 \( 1 - 1.68e4T + 8.44e8T^{2} \)
67 \( 1 + 6.90e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.05e4T + 1.80e9T^{2} \)
73 \( 1 - 4.61e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.86e3T + 3.07e9T^{2} \)
83 \( 1 + 1.06e5iT - 3.93e9T^{2} \)
89 \( 1 - 3.82e4T + 5.58e9T^{2} \)
97 \( 1 + 9.07e4iT - 8.58e9T^{2} \)
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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

−10.39517928964836893865320810255, −9.637507729712962745567765660067, −8.447305801103476015690434017989, −7.61823496052975091198767610075, −6.97106407634353889680145130625, −5.67699069288877489946192342468, −4.98848668523624527462452557060, −3.47320725000390641609498941787, −2.02309345427244151345157185089, −0.49693264196249917101078616217, 0.956161116937960971483251790146, 2.39901170603740291380652862506, 3.58853858822148740622765455036, 4.62984818182026453273507792633, 5.49181893011265932673529755754, 7.09570462475967052752446574334, 7.965426565093626205164725018594, 9.179176024657416743947243469174, 10.02448381174800094047451753265, 10.60629255045802745762877269843

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