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Properties

Label 2-350-5.4-c5-0-22
Degree $2$
Conductor $350$
Sign $0.447 - 0.894i$
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 + 17i·3-s − 16·4-s − 68·6-s − 49i·7-s − 64i·8-s − 46·9-s − 715·11-s − 272i·12-s − 331i·13-s + 196·14-s + 256·16-s − 1.69e3i·17-s − 184i·18-s + 1.71e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.09i·3-s − 0.5·4-s − 0.771·6-s − 0.377i·7-s − 0.353i·8-s − 0.189·9-s − 1.78·11-s − 0.545i·12-s − 0.543i·13-s + 0.267·14-s + 0.250·16-s − 1.42i·17-s − 0.133i·18-s + 1.09·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
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: \((2,\ 350,\ (\ :5/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.588386381\)
\(L(\frac12)\) \(\approx\) \(1.588386381\)
\(L(\frac{7}{2})\) 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 \( 1 - 17iT - 243T^{2} \)
11 \( 1 + 715T + 1.61e5T^{2} \)
13 \( 1 + 331iT - 3.71e5T^{2} \)
17 \( 1 + 1.69e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.71e3T + 2.47e6T^{2} \)
23 \( 1 - 3.95e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.57e3T + 2.05e7T^{2} \)
31 \( 1 - 6.75e3T + 2.86e7T^{2} \)
37 \( 1 + 1.65e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.88e4T + 1.15e8T^{2} \)
43 \( 1 + 2.25e3iT - 1.47e8T^{2} \)
47 \( 1 + 537iT - 2.29e8T^{2} \)
53 \( 1 - 1.09e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.59e4T + 7.14e8T^{2} \)
61 \( 1 - 3.91e4T + 8.44e8T^{2} \)
67 \( 1 - 4.41e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.18e4T + 1.80e9T^{2} \)
73 \( 1 - 5.01e3iT - 2.07e9T^{2} \)
79 \( 1 - 2.79e4T + 3.07e9T^{2} \)
83 \( 1 + 3.76e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.72e4T + 5.58e9T^{2} \)
97 \( 1 + 6.31e4iT - 8.58e9T^{2} \)
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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

−10.57752443192637267844597930175, −9.799375259608524416022015197766, −9.176747166514246907711093275198, −7.67772656745840568474321369764, −7.39442305784458136179207890557, −5.51441757194966960079643938234, −5.17148082133071264803394222048, −3.94805097309223833837832440191, −2.81557602060556826075794594913, −0.58102727409788131478990997147, 0.800794083188094380774345139492, 2.01283574758085709764277292845, 2.85038191261791535347148887050, 4.43241809883464917781612230318, 5.62237699697275029917362230310, 6.68627590212716470361602700298, 7.903656042881116029370356597644, 8.381581213438818775333932740666, 9.789823787617255118432247411140, 10.54164765101383559841887780534

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