↑

Properties

Label 2-935-935.274-c0-0-1
Degree $2$
Conductor $935$
Sign $-0.516 - 0.856i$
Analytic cond. $0.466625$
Root an. cond. $0.683100$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.17 + 1.17i)2-s + 1.76i·4-s + (0.382 + 0.923i)5-s + (−0.425 + 1.02i)7-s + (−0.899 + 0.899i)8-s + (0.707 − 0.707i)9-s + (−0.636 + 1.53i)10-s + (−0.923 − 0.382i)11-s − 1.96i·13-s + (−1.70 + 0.707i)14-s − 0.351·16-s + (−0.980 − 0.195i)17-s + 1.66·18-s + (−1.63 + 0.675i)20-s + (−0.636 − 1.53i)22-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)2-s + 1.76i·4-s + (0.382 + 0.923i)5-s + (−0.425 + 1.02i)7-s + (−0.899 + 0.899i)8-s + (0.707 − 0.707i)9-s + (−0.636 + 1.53i)10-s + (−0.923 − 0.382i)11-s − 1.96i·13-s + (−1.70 + 0.707i)14-s − 0.351·16-s + (−0.980 − 0.195i)17-s + 1.66·18-s + (−1.63 + 0.675i)20-s + (−0.636 − 1.53i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $-0.516 - 0.856i$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ -0.516 - 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.820685146\)
\(L(\frac12)\) \(\approx\) \(1.820685146\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (0.980 + 0.195i)T \)
good2 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + 1.96iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.149 - 0.360i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
89 \( 1 + 1.84iT - T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.42460389018029969938842356087, −9.800495148971700071651438312425, −8.525005176167038062074854110067, −7.74298377359141606006781761308, −6.87404925156132593108686195833, −6.06618651004489711535855782516, −5.66025979605600292837646403607, −4.60842089639246932197771362445, −3.24371284872052177611674029343, −2.75030722863826068481841560735, 1.54930350855381957802519194816, 2.33714376125482916054723212878, 3.90468204167750380976720417922, 4.54849604078199492519273265482, 5.01401775692909777061892391745, 6.37071412750514893221841003228, 7.23212386538904148130095044983, 8.484290608222104067158369411964, 9.589872071223483388871674226641, 10.21716187909897330129260061585

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